Some useful notes on the Real Life physics of objects this size:
In response to letters prompted by his original paper, Freeman Dyson replied: "A solid shell or ring surrounding a star is mechanically impossible. The form of 'biosphere' which I envisaged consists of a loose collection or swarm of objects travelling on independent orbits around the star." The evolution of the term "Dyson Sphere" is an example of Memetic Mutation. Dyson himself referred to his idea as a "shell" or "swarm", and his use of "biosphere" was in the ecological, rather than any geometric sense. But then the "sphere" misconception and Rule of Cool caught up with him.
Many ideas have the Dyson Sphere be 1 astronomical unit (AU) in radius, since this is the average distance from the Earth to the Sun and the intention is for the entire sphere to be at temperatures comfortable for Earth life. However, a smaller sphere is certainly possible: this would just require advanced cooling technology and/or most of the sphere not being used for habitation. This would also reduce the amount of material and energy needed to construct the sphere, e.g. a Dyson Sphere with a radius equal to Mercury's orbit (0.39 AU on average) would need a tenth of the material.
The surface gravity of the outside of a stereotypical 1-AU solid shell is likely to be negligible. Gravitational acceleration due to the Sun out there is less than 1/1000 g. Calculus and physics (the Shell Theorem) tell us that the gravitational effect of any spherical shell that we're outside of is equivalent to that of the same mass as a point source at the shell's center; we can assume that the sphere doesn't have mass orders of magnitude more than the Sun, considering that it has to be made of locally available materials and the Sun weighs much more than everything else nearby combined, so don't expect much. Also, increasing the mass of the star increases its brightness, so the sphere would be larger as well. However, it might be possible for a Sufficiently Advanced Alien to build a smaller, 'hot' Sphere, in which case the gravity would be higher.
The surface gravity on the inside of a 1-AU solid shell is also negligible. Newton's shell theorem predicts that the gravity anywhere inside a hollow sphere would actually be zero, so that the main gravitational force acting on a body at the inside surface would be the star at the center, which as stated above is not large. Spinning such an object would allow for near-normal gravity at the surface, but as was discovered by Larry Niven, the strength-to-weight ratio of the material required to withstand 1g of centrifugal force across this scale would be something around 1000 times that of steel. Of course, Speculative Fiction writers are allowed to handwave it as 'a network of gravity generators', Functional Magic, Unobtanium, etc. But if the solid shell idea is abandoned, it's possible to have a structure that spins "in place" (rather than around the Sun) to generate Earth-level gravity with regular materials. This is the idea behind the topopolis, essentially a cylinder that spins around its own axis while being so long it loops around the Sun one or more times.
A Klemperer Rosette is a group of objects set to orbit together in a rotating pattern. Technically this may count if it's big enough to be considered a small solar system, and the objects were placed there artificially.
A solid Dyson Sphere would not be stable around its star without some form of correctional thruster system or Applied Phlebotinum to keep everything bolted in place. Harder settings may couple the star to the mass streams of active supports holding up the sphere.
A point which is often ignored is that a solid shell will not only have eternal day, but also, instead of reflecting sunlight into space, will reflect it upon the sphere itself. This means that a shell around a Sun sized star will have to be not 300 million km across, but more likely about twice as much in order to maintain reasonable temperatures or using materials either with high efficiency in the production of energy (if the interior wall is dedicated exclusively to energy production) or with a high thermal conductivity coupled with an interior wall with high absorption of radiation. A reasonable partial solution to this is for a star-faring society (and any society able to build a Dyson Shell is likely to be star-faring at least potentially) to select a smaller, cooler star (say a class M) and build their Shell around it, letting it heat up to hotter intensities due to reflection.
Assuming that a solid shell Dyson sphere could be made to spin and produce Earth equivalent gravity, that 1G value of gravity would only exist at the equator (and arguably the tropic latitude regions of the sphere. As one moves to the hemispherical regions, the gravity reduces until there is zero gravity at or near the axis regions. This is one of the reasons that the Niven ring and topopolis are more practical. Much of the inner region of a fully enclosed Dyson sphere would be superfluous and uninhabitable. However, the uninhabitable portions of the inner surface could still be used for energy collection (which is closer to Dyson's original concept anyway).
A Dyson sphere could be used as a computational node for a single massively powerful supercomputer, presumably either a segment of, or the entire physical being of, some kind of massively advanced Artificial Intelligence entity, presumably of the post-Singularity kind. Such an entity would be called a "Matrioshka brain" named after Russian Matrioshka dolls, because it may consist of layers of Dyson shells within each other, the inner layer being devoted to energy harvesting while the outer structures are devoted to computation.
was in the ecological, rather than any geometric sense. But then the "sphere" misconception and Rule of Cool caught up with him.