I want to set a story within a zero gravity environment wherein creatures can live, breath the air, fly and flourish. The environment should supply all the things living creatures (as we know them) need to survive — except for a planet itself.
Can such an environment exist anywhere in nature?
Physicist and aerospace engineer Robert L. Forward was credited by Larry Niven with calculating the parameters that would allow a habitable gas torus to be stable in the short term (which in astronomic terms would be sufficiently long), and this was the setting of The Smoke Ring.
The gas torus is made up of a very large (about 9/10th if memory serves) volume where the pressure is gradually increasing and a central torus, still hundreds of kilometers across, where the pressure is suitable to human life.
Within this area, there is no real gravity except for tidal forces, and those only on large floating vegetable life-forms called integral trees. This volume (many times the habitable volume of Earth) seems to fit your requirements.
We can get a different, and still "natural" scenario by using a water moon. Provided that there's some way of keeping it liquid and preventing it from dissipating in space, of course; much the same problem as the Smoke Ring.
The moon itself would have a gravity and would collapse in a spherical surface, and it might have enough floating impurities to form a "peel" reducing water loss (they would also stop sunlight, not that much sunlight would get in the depths).
Unfortunately this scenario too is not original - this is from Iain M. Banks' The Algebraist:
I was born in a water moon. Some people, especially its inhabitants, called it a planet, but as it was only a little over two hundred kilometres in diameter, 'moon' seems the more accurate term. The moon was made entirely of water [...]
If it had been much bigger the moon would have had a core of ice, for water, though supposedly incompressible, is not entirely so, and will change under extremes of pressure to become ice. (If you are used to living on a planet where ice floats on the surface of water, this seems odd and even wrong, but nevertheless it is the case.) The moon was not quite of a size for an ice core to form, and therefore one could, if one was sufficiently hardy, and adequately proof against the water pressure, make one's way down, through the increasing weight of water above, to the very centre of the moon.
Where a strange thing happened.
For here, at the very centre of this watery globe, there seemed to be no gravity. There was colossal pressure, certainly, pressing in from every side, but one was in effect weightless (on the outside of a planet, moon or other body, watery or not, one is always being pulled towards its centre; once at its centre one is being pulled equally in all directions)...
I once read a fragment of fan-fiction possibly inspired by the Heechee's black hole refuge, where an ancient but reclusive K-2 race was able to create a hollowed out environment at the center of its home star. Then they artificially slowed down the star's reactions to ensure that it would live enormously longer than normal; finally, they settled its core, protected from the Universe by several hundred thousand kilometers of incandescent gases and intending to see their civilization last for some billion years.
In the core, again you would have no gravity; hollow stars definitely are not natural, though. But if the original civilization built sturdily enough, and yet died of boredom some million years later, leaving behind a self-sustaining environment where evolution could have a chance [which requires environment changes and/or a careful setup], that would allow for another race to appear, unaware of living in an artificial environment.
You could achieve a very similar zero-gravity effect with an atmosphere in which your creatures are neutrally buoyant. Jellyfish live in the water the way you propose.
You could have a thick atmosphere, possibly composed in part of denser gases. If the mass of my own body's volume is the same as the mass of an equivalent volume of atmosphere, I will be weightless. As opposed to a true zero-G enivroment there will still be things that fall and things that sink.
For zero gravity, you'll need to be far from any celestial body, so you want to be somewhere in empty space. To keep an atmosphere without gravity, you need an enclosed space. A simple solution is a large spaceship that doesn't have artificial gravity. You can make your spaceship as large as your plot requires, without needing it to be as large as a Dyson sphere, which requires solar system scale construction. Basically, you've got an atmosphere-filled cube/sphere/cylinder that drifts endlessly through space.
Rocheworlds
A Rocheworld is a pair of equally sized planets that orbit so closely around there common center of mass that their atmospheres touch. At that point of contact the gravitational pull from the two planets cancel. This would give you a microgravity environment. This is the L1 point of the two worlds and thus would have the gravitational gradients you would expect at L1. This means nothing would be stable there, denser then air organisms would fall toward the worlds, and lighter than air organisms would be pushed away from the center of the envelope. But these are minuscule forces that an organism with any level of activity should be able to overcome. It does mean floating islands and masses of inert vegetation are probably out.
On whether this configuration of worlds is possible, this question addresses sharing an atmosphere. To summarize, if the pull of the worlds on each other is to great that they tear themselves apart. There is a very slim margin in which two terrestrial worlds are close enough to share atmosphere but not destroy each other. But we don't care about the surface of the worlds so I think the better option would be two gas giants. Of note is that at the far end in this direction, stars that share an atmosphere, there are observed cases of their existence.
Speculative Environment
As for the biology, this is how I envision the ecosystem working. The amount of sunlight reaching the area of micro gravity is small due to the worlds eclipsing it on either end. You could have the orbital plane of the planets perpendicular to the plane of the ecliptic which would allow more light but I feel would reduce stability in an already tenuous system. Putting the worlds closer to their parent star would also hurt stability. My solution is to cover the gas giants with photosynthetic algae. The high gravity on the worlds themselves will preclude the development of complex life which leaves the algae without any predators so they can flourish. Air currents pass these algae across the bridge between worlds, possibly even concentrating the nutrients. This gives a food and oxygen source for the complex life in the microgravity environment in between the worlds.
As mentioned earlier, no organism here could be inert, every living thing that can't survive the high gravity of the worlds below, or the vacuum to the sides, would need to constantly be making micro adjustments to keep itself near L1. This could lead to a king of the hill type power struggle, where organisms are vying to be closest to L1.
Another possible plot point is the storms in such a system. I don't know exactly what they would be like, but I am sure they would be interesting.
Note: Rocheworlds have appeared in fiction, most notably Robert L. Forward's series which coined the term. Though I am unaware of any cases where the inhabitable micro gravity environment has been a key point. In part because in fiction the pair of worlds are usually terrestrial making the bridge between them much smaller in volume.
