I was thinking of writing a story and had the idea of having a society exist on the core of a gas planet. The core would be similar to earth with a breathable atmosphere similar to ours above it but still have the outer atmosphere of hydrogen and helium above it as these two gases are lighter than the mainly oxygen and nitrogen of the inner atmosphere and so the breathable atmosphere would sink down and allow the lifeforms on the planet's core to live. I was just wondering if this is possible at all in regards to science and its limitations or if this could never happen in real life.
First off, "breathable atmosphere sinking to the bottom" is not a valid assumption because gas giants are solid on the inside, due to the pressure as Thucydides pointed out.
Also, not only does Jupiter radiate more heat than it gets, Jupiter's core is hotter than the surface of the sun.
Gas giants are not big planets with a huge atmosphere, they are an almost failed star. It is the enormous pressure inside a Gas Giant that lets hydrogen to be found in metallic form. Liquid Metallic Hydrogen is some sort of degenerated matter and the required pressure to form such type of Hydrogen is 25 Gpa($3.5\times 10^6$ psi). At that kind of pressure Oxygen becomes solid, so no breathable atmosphere made of oxygen in the deep layers of a Type-I Gas Giants.
But how about the other types of gas giants? Turns out that are five of them in the Surdasky's Gas Giant Classification. In our solar system, we only have Type-I, which is the coldest. Type IV and V are what we call Hot Jupiters, to much close to their parent star and to hot for almost anything.
There's also the fact that Liquid Metallic Hydrogen is really conductive. It's responsible for Jupiter's magnetosphere which is fourteen times strong that ours on Earth. And not a good factor for life.
So, I guess that covers all basic facts of why not life can prosper inside a Gas Giant. The environment of and Gas Giant is simply to harsh to anything build up and later develop as life. Pressures, temperatures and constant electric currents also make impossible to life came from another place and make the Gas Giant his home. At least not in his core.
Probably they will have more luck setting up a station in the atmosphere of a Type II, that are made of water vapor.
It might be possible, depending on the size.
Per this question on the astronomy exchange, smaller gas planets can exist. The minimum density it would need would be is related to it's distance from the star it orbits though. As mentioned in Zxu's answer, you wouldn't be able to pull off a gas giant due to issues with pressure.
Alternatively, you could also have a planet with a solid core, but a tall and more opaque atmosphere which might give the appearance of a gas planet. You still wouldn't be able to get too large for the same reason. You also didn't specify if the inhabitants were on solid ground or not, so this may or may not be an option.