A Dyson sphere in the solar system, with a radius of one AU would have a surface area of at least 2.72e17 km^{2}, around 600 million times the surface area of the Earth. The sun has a energy output of around 4e26 W, of which most would be available to do useful work.
The original proposal simply assumed there would be enough solar collectors around the star to absorb the starlight, not that they would form a continuous shell. Rather, the shell would consist of independently orbiting structures, around a million kilometres thick and containing more than 1e5 objects.
Various science fiction authors seem to have misinterpreted the concept to mean a solid shell enclosing the star, usually having an inhabitable surface on the inside, and this idea was so compelling that it has been the main use of the term in science fiction. The earliest appearance of this version seems to be Robert Silverberg's novel Across a Billion Years.
A third kind of shell would be very thin and nonrotating, held up by the radiation pressure of the sun. It would consist of statites (see below, in the section about stability). Essentially it is a "dyson bubble", where reflecting sails reflect light onto collectors for use in external habitats. Its mass would be very smalll, on the order of a small moon or large asteroid.
In the following I will call solid Dyson spheres Type II or dyson shells and independently orbiting spheres Type I.
Dyson is a fellow of the Royal Society, a member of the U.S. National Academy of Sciences, a corresponding member of the Bavarian Academy of Sciences, a honorary fellow of Trinity College and an Associé Etranger de l'Académie des Sciences. He is president of the the SSI (Space Studies Institute).
Among his numerous awards and honors, Dyson received the Oersted Medal from the American Association of Physics Teachers, the Phi Beta Kappa Award in Science for Infinite in All Directions, the National Books Critics Circle Award for nonfiction, the 1981 Wolf Prize in physics, the Lewis Thomas Prize and many other honors.
Books by Dyson:
As the aeons advanced, hundreds of thousands of worldlets were constructed, all of this type, but gradually increasing in size and complexity. Many a star without natural planets came to be surrounded by concentric rings of artificial worlds. In some cases the inner rings contained scores, the outer rings thousands of globes adapted to life at some particular distance from the sun. Great diversity, both physical and mental, would distinguish worlds even of the same ring.Stapledon, in turn, may have got the idea from J. D. Bernal, who also influenced Dyson directly. Bernal describes in The World, the Flesh, and the Devil spherical space colonies:
Imagine a spherical shell ten miles or so in diameter, made of the lightest materials and mostly hollow; for this purpose the new molecular materials would be admirably suited. Owing to the absence of gravitation its construction would not be an engineering feat of any magnitude. The source of the material out of which this would be made would only be in small part drawn from the earth; for the great bulk of the structure would be made out of the substance of one or more smaller asteroids, rings of Saturn or other planetary detritus. The initial stages of construction are the most difficult to imagine. They will probably consist of attaching an asteroid of some hundred yards or so diameter to a space vessel, hollowing it out and using the removed material to build the first protective shell. Afterwards the shell could be reworked, bit by bit, using elaborated and more suitable substances and at the same time increasing its size by diminishing its thickness. The globe would fulfil all the functions by which our earth manages to support life. In default of a gravitational field it has, perforce, to keep its atmosphere and the greater portion of its life inside; but as all its nourishment comes in the form of energy through its outer surface it would be forced to resemble on the whole an enormously complicated singlecelled plant.According to Stefan E. Jones <stefanj@io.com>;, Raymond Z. Gallun, an american SF author may have come up with a similar concept independently.  
A star is essentially an immense reservoir of energy which is being dissipated as rapidly as its bulk will allow. It may be that, in the future, man will have no use for energy and be indifferent to stars except as spectacles, but if (and this seems more probable) energy is still needed, the stars cannot be allowed to continue to in their old way, but will be turned into efficient heat engines. The second law of thermodynamics, as Jeans delights in pointing out to us, will ultimately bring this universe to an inglorious close, may perhaps always remain the final factor. But by intelligent organization the life of the universe could probably be prolonged to many millions of millions of times what it would be without organization. Besides, we are still too close to the birth of the universe to be certain about its death.
Even if cheap and efficient fusion power can be developed, eventually the waste heat has to be radiated away by a Dyson spherelike cooling system.
Other proposed uses have been for security (although it is hard to hide the infrared emissions; energy could be radiated away in certain directions, but thermodynamics places some limits on it), or just for the fun of it (if you have a sufficiently advanced technology megaengineering could become a hobby activity; after all, ordinary people today perform engineering or crafting feats far beyond the imagination of previous eras).
This would correspond to an infrared wavelength of l = 2.8978e3 / T m (assuming a blackbody sphere) which for reasonable sizes lies in the infrared. Dyson predicted the peak of the radiation at ten micrometers.
The sky would be filled with the surface of the sphere, giving the impression of a huge bowl over a flat earth, covered with clouds, continents and oceans although for a real Dyson shell these would have to be immense to be noticeable. The angular size of an object at a distance d and diameter l is 2arctan(l/2d). For an object of diameter 10,000 km (like the Earth) at a distance of a 100 million km (around 120 degrees away from the observer on the shell), the angular size would be around 10^{4} rad or 0.005 degrees, roughly the size of a pea 100 meters away.
It should be noted (as Richard Treitel has pointed out) that even a very dark surface will shine intensely, making the sky much brighter than on Earth. The albedo of Earth is around 0.37, so an interior with an earthlike environment would have a sky where each patch reflects a noticeable fraction of the sunlight.
In a Type I dyson sphere roughly the same things would be seen: a plane wall of orbital habitats, solar collectors and whatnot stretching away into what looks like infinity (although here the curvature may become noticeable for observant viewers) and a hemispherical bowl covering the rest of the sky, centered around the sun. Solar collectors would have a very low albedo, but it is still likely that the interior will be very bright.
Another version would be based on statites (this is probably due to Robert L. Forward): each solar collector will also be a solar sail, and hover without orbiting above the sun, held up by light pressure. By adjusting the sail area statites can move in and out, and by adjusting their angle they can move away if needed. Traffic control may be a problem, but can likely be handled in various ways, for example by local flight control centres or automatic systems based on flocking behaviour.
The force on a statite would be F = L/(4pc r^{2})  GMm/r^{2}, where L is the total luminosity of the sun (3.9e26 W), M is the mass of the sun, m is the density of the statite, r the distance to the sun and c is the speed of light. To remain in balance, the statite will have to have the density
A rigid dyson sphere is not stable, since there is no net attraction between a spherical shell and a point mass inside. If the shell is pushed slightly, for example by a meteor hit, the shell will gradually drift off and eventually hit the star. This is a classic problem in elementary mechanics and is usually solved in introductory textbooks.
Divide the shell into narrow rings. Let R be the radius of the shell, t its thickness (t << R). The ring at angle q, which subtends an angle dtheta, has a circumference 2pR sin theta, width R dq an thickness t, which gives it a volume of
dV = 2pR^{2}t sin q d qand a mass of (M/2) sin q dq where r is the density of the shell.
Each part of the ring is the same distance r´ from m, and by symmetry the force from the ring is directed along the axis with no transversal component. Since the angle alpha between the force vector and the line of centres is the same for all sections of the ring, the force components along the line of centres add to give
dF = Gmr dV cos a / r´^{2}for the whole ring. This is then integrated: F = int (G mr dV/r´^{2}) cos a. By expressing cos a as a function of polar angle we get:
F = [GMm/2] ò _{0}^{p} ( (r  R cos q) sin q dq)/(r^{2} + R^{2}  2 r R cos q)^{2}/3Through the substitution u = rRcos(q), du = Rsin q dq we get:
(where ò _{0}^{p} is the integral from 0 to p).
F = [GMm/2R] ò _{{rR}}^{{r+R}} (u du) / (R^{2}  r^{2} +2ru)^{3/2}which is a standard integral resulting in:
F = (GMm/2R)(1/2r^{2})[sqrt(R^{2}r^{2}+2ru)(r^{2}R^{2})/sqrt(R^{2} r^{2}+2ru)]_{{rR}}^{{r+R}}For r < R we get:
F = (GMm/4Rr^{2}){(R+r)(Rr)(r^{2}R^{2})(1/(R+r)1/(Rr))} = 0
Any sphere about a gravitating body can be analysed into two hemispheres joined at a seam. The contribution of a small section To the force on the seam isg(ravity)*d(ensity)*t(hickness)*A(rea)*cos(angle).The integral of A cos(angle) is pR^{2}.So the total force is g d t pR^{2}. Which is independent of distance, neatly enough.
The area resisting the force is 2pRt.
Thus, the pressure is g d R/2; this can be translated into a cylindrical tower of a given height on Earth. If that tower built of that material can stand, then the compression strain is not too great.
At 1 AU, that comes to 2(p*AU/YR)^{2}, or  by my calculations  in the neighborhood of 80 to 90 THOUSAND kilometers high.
The tendncy to buckle, moreover, is another problem.
The only ways to make a rigid Dyson shell habitable on the inside would be either to provide it with some sort of antigravity (which is unlikely) or to rotate it, which would make only the equatorial band habitable unless the interior was terraced. A rotating dyson sphere would be under immense strains; see the section about the ringworld for a simple calculation. Niven pointed out that if you want to spin a Dyson sphere, it is better to build it like a film canister for reasons of structural strength, and then you have a Ringworld.
It has been suggested that one could live on the outside of the sphere, especially if the interior star is rather cool; it appears that a terrestrial environment is possible around M stars just at the end of the main sequence. Erik Max Francis gives the following derivation of this kind of sphere:
First, know the luminositymass relation for main sequence stars:
L = kM^{n},where k is a constant of proportionality and n is between 3.5 and 4.0. (k depends on the choice of nu, obviously.) You can find the constant k, given n, based on the fact that the Sun has a luminosity of 3.83
Second, know the gravitational acceleration:
g = GM/R^{2}.Third, the blackbody power law (we're approximating the star as a blackbody, which isn't too bad of an approximation):
L = esAT^{4}.Knowing these factors, you can combine them to get an equation which relates the mass of the star to the desired temperature and gravity of the sphere:
Substituting ideal conditions (g
= 9.81 m/s^{2}, T = 300 K), you find that M
must be between 0.054 and 0.079 masses solar (the variance is dependent
on the variance in the exponent in the massluminosity relation). The end
of the main sequence is at about 0.08 masses solar, for comparison.
This would produce spheres with a radius of 0.00570.0069 AU (852,720  1,032,240 km).
It might also be possible to have a biosphere between two dyson spheres (this is used in Baxter's The Time Ships).
It should be noted that there would be no auroras in a dyson shell, since there is no magnetic field. This also would also mean that more radiation would reach the ground from the sun since it cannot naturally be deflected (although one could imagine megaengineering systems to provide an artificial magnetic field).
A rigid dyson shell would require superstrong materials, and its construction is complicated since half a shell is unstable. One could conceive of some dramatic capping process, where a number of previously freely orbiting structural components at the same time moved inwards to lock together into a shell (for example twenty spherical triangles). This would require tremendous precision, but since supertechnology is already assumed for building a rigid shell, it seems almost trivial. As somebody put it, if you can build a dyson shell you don't need it.
If one assumes that all elements heavier than helium are usable (a slight
exaggeration), then the inner planets are completely usable, as is the
asteroid belt.

(1E24 kg) 

Mercury  0.33022 
Venus  4.8690 
Earth  5.8742 
Moon  0.0735 
Mars  0.64191 
Asteroids  ~0.002 
Sum:  11.78733 e24 kg 
It is a bit more uncertain how much of the outer planets is usable. Jupiter and Saturn mainly consist of hydrogen and helium, with around 0.1% of other material. Jupiter is assumed to have a rock core massing around 1015 times the Earth, and Saturn probably contains a smaller core massing around 3 times the Earth. Uranus and Neptunus seem to be mainly rock and ice, with around 15% hydrogen, so a rough estimate would be around 5070% usable mass. Pluto seems to be around 80% usable.

(1E24 kg) 
(rough estimate) 

Jupiter  1898.8  ~58 
Saturn  568.41  ~17 
Uranus  86.967  ~43 
Neptune  102.85  ~51 
Pluto  0.0129  ~0.01 
Kuiper belt objects  ~0.02  ~0.016 
Sum:  2657.06 e24 kg  ~170 
(these tables based on information from Physics and Chemistry
of the Solar System by John S. Lewis and The
Nine Planets by Bill Arnett)
The inner system contains enough usable material for a dyson sphere. If one assumes a 1 AU radius, there will be around 42 kg/m^{2} of the sphere. This is probably far too little to build a massive Type II dyson sphere, but probably enough to build a Type I dyson sphere where mass is concentrated into habitats and most of the surface is solar sails and receivers, which can presumably be made quite thin.
With the extra material from the outer system, we get around 600 kg/m^{2}, which is enough for a quite heavy sphere (if it was all iron, it would be around 8 centimeters thick, and if it was all diamond around 20 centimeters).
A Type III shell, a "dyson bubble", would have a very low mass. Since its density is independent of the radius (see the stability section), its mass would scale as r^{2}. For an 1 AU bubble, the total mass needed would be around 2.17e20 kg, around the mass of Pallas.
T = [E / (4ph s r^{2})]^{1/4}where h is the emissivity (=1 for a blackbody), s the constant of StefanBolzman's law (5.67032e8 Wm^{2}K^{4})and E the total energy output of the star measured in watts.
In theory, if eta is very low the interior of the sphere could become as hot as desired, but this is unlikely since the material of the sphere would start to melt or evaporate if the temperature moved above 20003000°K or so. And if the surface of the star became hot enough, the outer parts of the star would expand and a new thermal equilibrium set in with less internal energy production. If the sphere was a perfect energy container the star would eventually expand until its fusion processeses ended; if the temperature was lowered (by energy use) fusion would resume until an equilibrium was reached  a bottled star.
It should be noted that at 1 AU, the energy flux is around 1.4e3 W/m^{2}, which calculates as around 395°K, or 122°C if the sphere is a blackbody. This is a bit too hot for an earthlike biosphere (Earth is cooled by its rotation, which effectively halves the energy flux, and its spherical shape, that lowers it further), and a dyson shell need some rather impressive cooling to work.
The radius of the smallest passively radiating shell with thermal tolerance T_{max} is
r_{smallest} = sqr(E / (4phs T_{max}^{4}))Diamond can stand around 4000°K; putting 4000°K into the equation, we get 1.48e9 meters, or around 1.4 million kilometers. For 1000°K we get a radius of 2.37e10 meters, or around 23 million kilometers. This is roughly 2 and 32 solar radii respectively. With active cooling the shell can be made much smaller.
Just as with a Type II Dyson sphere the internal stresses would require an immensely strong material (Niven uses the invented material scrith, a greyish translucent material with strength on the order of the nuclear binding strength). The stress is
F = r r g [N/m]where r is the weight of the ringworld per square meter (kg/m^{2}) and g is the surface acceleration and r is the radius. For the ringworld g was close to earthly, a radius of around 1 AU and there was at least a kilometre of surface material of approximately eartlike density. This would provide a stress on the order of 1e181e19 N/m.
The ringworlds instability is also (in)famous. It is not neutrally stable like a dyson sphere, but dynamically unstable  a small disturbance (such as the inhomogenities in the solar wind or meteor strikes) will grow gradually, and the ringworld would gradually loose its centeredness until it runs into its sun (the ringworld is transversely stable, if the ring is perturbed along its axis it will oscillate around the equilibrium position). See Erik Max Francis' page abour ringworld stability for an easy derivation. Niven solves this problem in Ringworld Engineers by placing ramjets along the edges, forming an active stabilisation system.
A related idea to ringworlds is Ian Bank's orbitals. An orbital is a small ringworld orbiting the sun (instead of encircling it; this circumvents the instability), with a rotation period of 24 hours and earthlike gravity due to the spin. Its size would be
Assume that you wish to have a large living volume in the form of a shell. The shell contains air, people, houses, furniture, etc. that averages 10 kg/m^{3}. If you fill the interior of the shell with hydrogen (the lightest gas) at room temperature, it will assume a distribution based on selfgravity. If the pressure at the inner shell boundary, where the hydrogen and living space meet, is 1 atmosphere, then there is a largest size you can build such a structure before the selfgravity of the hydrogen starts to make it smaller.
The living space has a thickness of 2400 km if you assume that the outer surface is at a pressure equal to that at 3000 m (10,000 ft) above sealevel on Earth. Such a bubbleworld would have about 5 million times the useable living volume of the Earth. The atmosphere is held in by a cap of material (such as 500 meters of ironnickel) so as to balance the gas pressure from below. The entire structure is in pressure equilibrium, so it requires no particular structural strength.
A rotating bubbleworld would be a flattened ellipsoid and could be several times larger, but determining the shape is more complicated than the nonrotating spherical case.
The living volume of the Bubbleworld would be in a 0.001 to 0.01 gee environment, making unusual architecture and humanpowered flight possible. The entire bubbleworld would mass about 3 Earths in mass.
Paul Birch "A Visit to SupraJupiter" Analog December 1992
The problem with this is that the amount of energy that can be extracted from the radiation depends on the difference in temperature on the two sides of the shell, and inside the star this will be rather low, while outside the star the difference will essentially be between the shell temperature and the cosmic background radiation. But it should be noted that if neutrinous can be captured, they would provide a kind of temperature differential that could be used (since the sun is almost transparent to them).
The result would be a sphere of carefully aligned orbiting matter just larger than its Schwartzhild radius, with a black hole in the middle. Instead of relying on energy from stellar fusion, matter could be fed into the black hole, releasing energy which would be used by a surrounding "galactic dyson sphere". The total size would be on the order of a few lightmonths.
Using black holes for energy production can also be done using smaller dyson spheres. A very small black hole will radiate intense Hawking radiation, quickly loosing its mass. If an equal amount of mass is swallowed (for example in the form of garbage) the hole will remain stable, and convert matter into energy which can be collected by the dyson sphere.
DATE: 1980 OBSERVER(S): WITTEBORN SITE: NASA  U OF A, MT. LEMMON INSTR. SIZE (M): 1.5 SEARCH FREQ.(MHz): 8.5 microns  13.5 microns FREQUENCY RESOL.(Hz): 1 micron OBJECTS: 20 STARS FLUX LIMITS (W/m**2): N MAGNITUDE EXCESS < 1.7 TOTAL HOURS: 50 REFERENCE: COMMENTS: Search for IR excess due to Dyson spheres around solar type stars. Target stars were chosen because too faint for spectral type. DATE: 1984 OBSERVER(S): SLYSH SITE: SATELLITE INSTR. SIZE (M): RADIOMETER SEARCH FREQ.(MHz): 37x10**3 FREQUENCY RESOL.(Hz): 4x10**8 OBJECTS: ALL SKY 3K BB FLUX LIMITS (W/m**2): T/T =< .01 TOTAL HOURS: 6000 REFERENCE: 27 COMMENTS: Lack of fluctuations in 3K background radiation on angular scales of 10**2 Strd. rules out optically thick Dyson spheres radiating more than 1 solar luminosity within 100 pc. DATE: 1987 OBSERVER(S): TARTER, KARDASHEV & SLYSH SITE: VLA INSTR. SIZE (M): 26 (9 ANTENNAS) SEARCH FREQ.(MHz): 1612.231 FREQUENCY RESOL.(Hz): 6105 OBJECTS: G357.31.3 FLUX LIMITS (W/m**2): TOTAL HOURS: 1 REFERENCE: COMMENTS: Remote observation (by VLA staff) of IRAS source near galactic center to determine if source could be nearby Dyson sphere. Source confirmed as OH/IR star.In short, none have been observed yet.
Kardashev, N. S., and Zhuravlev, V. I., SETI in Russia, paper presented at the IAA/COSPAR/IAF/NASA/AIAA symposium on SETI: A New Endeavor for Humankind, The World Space Congress, Washington, D.C., August 30, 1992. To appear in a special issue of Acta Astronautica.
Jugaku, J., and Nishimura, S., A Search for Dyson Spheres Around LateType Stars in the IRAS Catalog, in Bioastronomy: The Search for Extraterrestrial Life, J. Heidemann and M. J. Klein (Eds.), Lectures Notes in Physics 390, SpringerVerlag, 1991
An article about dyson spheres by Sarah Voigt:
Dyson
Spheres: A Primer
A miniFAQ, covering the basics (at SEDS): The Ultimate Biospheres
Illustration of a dyson sphere: http://www.setiquest.com/dyson.htm.
Picture of F. Dyson: http://www.setiquest.com/dyson2.htm.
Computer graphics of a Ringworld: http://www.rahul.net/rootbear/graphics/ringworld/index.html
Images of dyson spheres: http://www.algonet.se/~aleph/Trans/Tech/Megascale/dyson_page.html
The Ultimate Biospheres: http://seds.lpl.arizona.edu/nodes/NODEv4n310.html
Transhuman Technologies, Megascale section: http://www.thehub.com.au/~mitch/extro/mega.html
Megastructures in science fiction by Ross Smith: http://www.geocities.com/SiliconValley/Park/3699/sfmegastructures.html
Outside Dyson spheres by Erik Max Francis: http://www.alcyone.com/max/writing/essays/outsidedyson.html
Dyson, F. J., The Search for Extraterrestrial Technology, in Perspectives in Modern Physics (Essays in Honor of Hans Bethe), R. E. Marshak (Editor), John Wiley & Sons, New York, 1966
Marshall T. Savage: The Millennial Project (ISBN 0316771635). Describes a plausible spacecolonization scenario, involving the construction of a Type I dyson sphere.
Star Maker (1937) by Olaf Stapledon (An enthusiastic review)
The World is Round by Rothman
Larry Niven: Ringworld, Ringworld Engineers and Ringworld Throne
Lord Kalvan of Otherwhen by H. Beam Piper
"Relics" episode of Star Trek The New Generation (regarded as very bad by many sf lovers)
Cageworld 1: Search for the Sun, Cageworld 2: The Lost Worlds of Cronus, Cageworld 3: The Tyrant of Hades and Cageworld 4: StarSearch by Colin Kapp.
Orbitsville (1975), Orbitsville Departure (1983) and Orbitsville Judgement (1990) by Bob Shaw
Across a Billion Years by Robert Silverberg
Farthest Star (1975), Wall Around a Star (1983) by Frederik Pohl & Jack Williamson
The Time Ships by Stephen Baxter
The Wanderer by Fritz Leiber (1967?) mentions in passing that the light from most of the stars in the inhabited galaxy are dimmed by the density of habitats orbiting them. (David Lorenzo Duffy <dlduffy@welchlink.welch.jhu.edu>)
Dyson spheres need great big walls
To keep the world from spilling out
They make them out of buckyballs
And use gravitons for grout
Mister Skin < mrskin@mindspring.com>