I'm reading a book (Sun of Suns by Karl Schroeder) that the main location is a planet called Virga, which contains air, water, and floating chunks of rock, and has no or a very small amount of gravity. There is a main 'sun' at the center of the planet, which provides the heat for weather.
Could a 'planet' of this type exist?
A sun or a star is not possible to exist on this scale; to be as massive as a core of a planet, it's just not massive enough. But you didn't mention the size of it.
So if we put that aside, first of all there's no such thing as no gravity. Where there's mass there's gravity, and that gravity has to be strong enough to hold gas (atmosphere). And the rocks will have to sink into the core since they are the denser objects.
If however we compared this to an existing example, where the sun is in the center of the solar system and holding planets (floating chunks of rocks), there's still vacuum in between. Because at the distances these planets are from the sun, the sun's gravity isn't strong enough to hold gas. Where the sun's gravity is strong enough there's gas, and that ends as far as the outer atmosphere of the sun itself. Which mainly doesn't extend to the planets. Therefore it's not possible.
If you look in outer space, you'll see things like giant molecular clouds these clouds are not necessarily in equilibrium, so the factors that cause them to exist for a certain amount of time may be very complicated. E.g., there could be shock waves, star formation, ...
If the cloud is in thermal equilibrium, then the typical molecular speeds go like $mv^2\sim kT$, and escape velocity is given roughly by $v^2\sim\Phi$, where $\Phi$ is the gravitational potential. The result is that the maximum temperature of such a cloud is $T\sim m\Phi/k$. If you put in a typical numbers, you find that even for a body with gravity as strong as the moon's, it's not possible to have air and water (high $T$ and low $m$). But note that the result depends on the gravitational potential, not the gravitational field, so in theory this could work if the body is very large in linear dimensions. Also, it would be possible, for example, to give the moon a permament atmosphere of heavy molecules such as long-chain fluorocarbons, making it a shirtsleeve environment where all you'd need was an oxygen tank.
The WP article for Sun of Suns describes a fullerene shell holding the whole thing together.
