Moon of a Large Gas Giant
I think most physicists believe this to be entirely possible. A gas giant can get very large before it ignites to form a star. Some estimates are that it would take TEN Jupiter masses before nuclear fusion could ignite. So Jupiter is already huge, but you can get a lot bigger and still be a gas giant planet. A big gas giant can have very large moons: Ganymede is larger than Mercury or Pluto. Something the size of Mars or Earth could easily orbit a large gas giant.

As for radiation, the inhabitants of such a moon would be even better off than they are on Earth. Jupiter has a massive magnetic field that deflects a great deal of solar and interstellar radiation. And an earth-size moon could easily have its own strong magnetic field adding to that protection. Even Ganymede has its own magnetosphere.

As for warmth, it's true that Jupiter is quite a ways out in the solar system, and thus gets a tiny fraction of the solar radiation the Earth does. This means that Ganymede, Callisto, and Europa are all frozen solid on the surface. But we now know that jupiter-like gas giants don't need to be far from their star. Our planetary surveys have shown many "hot jupiters" orbiting nearby stars at many distances, from the baking zone of Mercury and Venus out to the goldilocks zone (liquid water) of Earth. It's entirely possible to have an inhabited gas giant moon at a comfortable range. Plus, the gas giant itself is likely radiating some heat. Jupiter does this: it puts out more heat into the universe than its taking in. We're not entirely sure why...we know it's not fusion going on, but must be some other process deep within the planet.

And while its true that Io is super hot due to jupiter's tidal forces heating it up, that wouldn't necessarily happen to any inhabited moon. It's all a function of the orbital distance.

Atmosphere is no problem for such a moon. Titan, for instance, has a very thick atmosphere with wind, rain, etc.