Jupiter is made up of a majority of hydrogen and helium gas, so there is no real surface to stand on. If you were to try and stand on the surface, you would sink until you reached a solid core at the center of the planet.

No, we could not stand on Jupiter. Jupiter has a solid core, but it's under so much pressure that the atoms of your body would be crushed into a metal were you to get that deep (note: this is not nuclear fusion; it's called electron degeneracy, and is similar to what happens inside of a white dwarf star ).