While flying and fighting in and around Jupiter's moons might be a lot of fun you'd definitely not want to get too close to Jupiter itself. If you did then it certainly would be game over because you'd never be able to escape its well known gravity. The same would apply to Saturn or any of the other gas giants.
Actually, that's no quite true.
Sure, Jupiter has a higher gravity than Earth, but it's "only" 2.5g. Saturn has a surface gravity similar to earth (1.065g). Uranus actually has a lower surface gravity compared to Earth at 0.886g.
If you are wondering why that is the case:
http://en.m.wikipedia.org/wiki/Surface_gravity
Now, the escape velocity to get away from these planets will need to be greater compared to earth but since we will be using the FSD for this normal physics doesn't really apply.