"Could" is a difficult word.
We should reiterate that anything could happen in a fantasy world. It could literally be made out of cheese if you wanted it to be. Naturalism is not the only route here.
However, if you do want to be naturalistic, then you have some pretty massive problems here.
A planet with no core
The core is what creates the magnetosphere. No core, no magnetosphere. Also, no plate tectonics, which probably means no carbon cycle, so probably no breathable atmosphere and little climate stability.
A planet with no metals
People are made out of metals. If you have no metals, you have to complete throw out earth biology and start from scratch. [the obvious one is calcium for bones, but small quantities of iron, magnesium, selenium and so forth are essential for cellular processes.
A gas core
What happens if your world has gas at the centre? Well, we need to distinguish two very different scenarios: hollow earth; and gas planet.
A hollow earth
In this scenario, the mantle is made of rock, but the core is made out of gas. What happens? Well, it could get quite complicated.
The first thing to say is that it's almost certainly impossible. You know what happens when you have a lot of rock sitting on top of gas? It falls down. The entire weight of the mantle would be pushing down on its inner surface, and nothing but hot air would be holding it up. No vault could be perfect enough to not allow some of the rock to fall - and then ALL of the rock would fall.
Now, it could theoretically be possible. If you make the gas hot enough and dense enough, the pressure of the gas attempting to escape could equal the weight of the rock pushing down on it. It's hard to imagine this working in practice, though - surely, even if it were globally true, there would be small areas where it wasn't true and grains of rock would fall. As rocks fell, the gas would expand into the gaps, and eventually it would find a way to burst out and the planet would explode. After all, our hot liquid rock in our planet periodically bursts out the top - but fortunately, liquid is a lot less explody than a pressurised gas!
The real physical killer, though, is how such a situation would arise. It's one thing if a god sets it up like that (they could make the inner surface of the mantle out of solid, impervious unobtainium, for a start). But in nature, you could never all all the rock converging simultaneously to trap exactly the right amount of gas in the middle - instead, like grains in a sieve, the rock would inevitably fall to the bottom while the gas floated to the top.
One fun thing about a hollow earth, though: inside the mantle, there is effectively no gravity!
Oh, and about hydrogen forming a magnetosphere: yes, if it's metallic hydrogen. Specifically, it needs to be liquid metallic hydrogen. Metallic hydrogen only exists at insane pressures (it's debated whether we've ever managed to produce any). Even the pressure at the centre of our Earth is not great enough to produce metallic hydrogen!
A gas planet
It is conceivable that life could exist in a gas planet. There can be a band in which liquid water is possible, and conceivably at reasonable temperatures and pressures - such a band does exist on Venus, for instance, which isn't a gas planet but does have a big atmosphere. So far as I'm aware such bands don't exist on our gas giants - high enough up to have normal pressure, and you're very, very cold; warm the place up and you get crushed. But it should theoretically be possible, I'd have thought?
Obviously, though, you wouldn't have any earthlike life. Clearly you couldn't have continents and oceans - as you can observe, rocks are heavier than air, so they would fall down (once you reached air that was dense and hot enough for continents to float on... well, then you'll have some survivability problems!).