The only weapon class I know of that's outright illegal to own is anti-aircraft missiles. That carries life imprisonment just for possession, with no violent intent. Because the government never wants to give up its air supremacy. This is why whenever you hear of feds concocting an international weapons conspiracy they always have to add anti-air bazookas to the charges because it's the only thing that actually can unequivocally be proven as illegal to own[0].
Basically everything else can be owned with an NFA tax stamp. Nuclear weapons my understanding is the difficulty is more with laws on handling the material than specifically owning one as a weapon, so I'm unsure those are even outright illegal either.
Explosives are actually one of the ones with looser restrictions. Even felons can own and re-instate their explosives rights, because bafflingly when congress de-funded the firearms rights restoration process for felons they forgot to do the one for explosives. Felons can also own and manufacture explosive black powder without scrutiny or paperwork, even ones intended to go in a black powder gun.