What about someone who will not be able to pay?

Follow the reasoning. He will remain liable.

See the Tur, who discusses this there. I brought several medieval authorities there from whom it seems that if he intends from the outset not to pay, then it is forbidden to save himself. But perhaps even according to their view, if that is not his intention and he simply will not be able to pay, then it is permitted and he remains liable—at least on the meta-halakhic level.

What do you mean by “he will remain liable”? My question is whether one may save himself with another person’s money even in a case where he will have no possibility of paying it back afterward, from the meta-halakhic perspective. For example, if a person needs an expensive operation to save his life, and he has the option of stealing the amount from someone, but since it is an astronomical sum, he knows he will never be able to repay it in the future. Is he allowed to do that?

So I already wrote: since the rationale is Tosafot on Sabbath 4—that is, Jewish law does not require someone to act in a way that has no chance of being obeyed—therefore in practice it is hard to rule like Rashi. We allow a person to save himself and remain liable. Therefore I think (not certain) that even if he has no way to pay, we still allow him to save himself and remain liable. Either he will raise money, or they will collect from his assets, and whatever they can manage to collect, they will take. Like any debtor.

Would he also be allowed to use violence in order to carry out the theft of the money? For example, to beat someone in order to get his money out of him (obviously without endangering his life), or to threaten him at gunpoint while knowing full well that he would never actually carry out the threat.

My intuition says no. Taking money can be repaid; bodily injury cannot. So too, according to the other medieval authorities, it seems that I am forbidden to injure a person in order to save myself, even though I am allowed to take his money.

But there are people for whom having their money taken from them—say all their money, for the sake of discussion—is harder than bodily injury. So why is that permitted while the other is not?

There is an explicit rabbinic midrash about that regarding “with all your might.” So it is hard for them—so what? That can be repaid, while bodily injury cannot.

But in a case where a person knows he will not be able to pay it back, it cannot be repaid.

[A simple opinion, plain as plain can be, is that this whole thing is a matter of a result-oriented categorical imperative: if one were to “allow” harming property rights (which as an independent principle is basically a lot of nonsense) in order to save, meaning that all people become a free insurance company for those in need of rescue, then overall suffering would increase more; and as much as possible, these matters are not left open to gradations.]

In principle it can be repaid. There will be a debt, and they will collect whatever they can manage to collect—from donations and charity, from money that may come to him later, etc.

But depriving the other person of the money can also lead to bodily harm to him—for example, if he himself needs the money for medical treatment. After all, an absurd situation could arise where there is a rich person and a poor person, and both need an expensive operation to save their lives. Suppose the rich person has enough money to cover the operation and no more. Then the poor person steals from the rich person in order to save himself, gets the operation, and the rich person is left to die.

Those are pathological cases. If it is known that this is the case, then of course it is forbidden.
Moreover, this whole law is talking about a situation where there is a concrete asset belonging to a specific person, and harming it will save me—like the haystacks in the case of King David. Just randomly robbing some person is problematic, because the obligation to save me rests on all people, not specifically on him. I think that is precisely why the state regulates treatment costs, in order to distribute the burden across the public as a whole.
After all, here we are discussing my view, according to which on the conceptual level it seems one should rule like Rashi, and the discussion is what to do in practice. But according to most medieval authorities, it is permitted by the basic law itself to steal in order to save oneself, unlike Rashi. According to them, am I also allowed to steal a kidney? It seems to me the halakhic decisors agree that I am not. In my view, most of them also distinguish between a specific asset designated for this and just randomly robbing a person. So these distinctions are made as a matter of Jewish law, and are not merely my own speculations within Rashi’s view.

It is written explicitly:

“Follow the money, and it shall you serve, and to it shall you cleave.”