Setting aside for a moment whether or not it's possible to find a gas that can be compressed to the density of a human without undergoing a phase change, suppose you are neutrally or nearly-neutrally bouyant in some fluid. The fluid has your density or higher.
Firstly, your lungs will struggle to move the fluid in order to "breath", since more force is required to shift a larger mass of fluid. That's a problem but it's not necessarily instantly fatal. Secondly, any significant "depth" of such fluid will have very high pressure on a planet, since there's a lot of weight of this stuff above you. Divers in water (approx same density as a human) can go to to a few hundred meters maximum in Earth gravity.
Hence, you need to look for micro-gravity situations (where everything "floats" irrespective of density), or else for environments where there isn't miles of fluid above you (so, not really an "atmosphere" worth the name). If we're not allowed a roof, then perhaps a small body with low gravity, and some kind of atmosphere-generator, would be sufficiently "floaty" even though really you are much denser than the atmosphere. You're sinking, just at very low acceleration. You'd have to think this is inefficient, though, since without a roof the atmosphere on a low-gravity body will be leaking into space.
As for whether a fluid you float in can even be a gas rather than a liquid: it doesn't seem likely that any gas could have a density close to that of a human, without being at such high pressure that a human can't survive.
Of course hand-wavium is good for a lot of engineering applications, but it doesn't qualify as "hard science". If you just assert the existence of a non-toxic gas that's 20% denser than water at standard temperature and pressure, and mix it with 20% oxygen, then you'd float in it and breathe it (albeit with increased difficulty compared to air). You'd also have to explain why the oxygen doesn't all float to the top. And for the reasons above to do with weight of fluid above you, the atmosphere would have to be a lot shallower than Earth's, or the gravity far weaker, or some combination of the two.
Now, if you're willing to consider a "liquid atmosphere", then you're in business. Just grow gills, use water (which will need to be oxygenated somehow) and don't go too deep into the "atmosphere" (or as it's more typically called, "ocean") ;-)