I'm sure I'm not the first to notice this, but stars at the bottom of the main sequence have the special property that their luminosities put their Habitable Zones at the same distance at which orbital acceleration is equal to 1g.
The reason this is interesting is that if you built an actual solid sphere with radius 1.5 solar radii around a 0.1 solar mass star then its effective temperature would be about the same as Earth's and so would its surface gravity. I'm not talking about just a Dyson-sphere style swarm of collectors or radiators, but a structurally rigid object.
Now these numbers are approximate -- I hope Jim Kasting doesn't catch me just scaling effective temperatures to calculate my "habitable zone" here, and I haven't done a rigorous mass-luminosity relation, nor have I considered the back reaction of the sphere on the star. But somewhere around 0.1 solar masses is about right for this special configuration.
This means that if you could build such a thing -- and I don't think you could, even just considering the mechanics of light-travel time and how stresses on the sphere would be balanced -- then you would have an object with 22,000 times the surface area of Earth, about its temperature, and about its surface gravity. Oh, and you'd have 10-4 solar luminosities of energy to work with, doing all sorts of work inside the shell. The waste heat from this work would warm your cities and your atmosphere (there's a lot of it) as you went about your life in eternal nighttime. The amount of material needed is not a problem: if you deconstructed the Earth and spread it out over this area, it would form a solid shell over 100 meters thick.
So, in contrast to Star Trek:TNG's imagining of having people living inside their huge star-spanning sphere (above; I guess you can do that when you have artificial gravity) you could very comfortably live on the outside, no antigrav required.... except for the whole gravity-crushes-the-impossibly-large-structure bit. And the gravity-means-it-quickly-drifts-into-the-star bit. Those would require some major Treknobabble to overcome.
So there is no reason to think that these could exist, but the formal existence of this solution to the problem, engineering aside, is not something I had noticed before.
[Image: art from the Star Trek: The Next Generation episode Dyson Sphere, fun but totally wrong episode based on a misinterpretation of the idea of a Dyson sphere]