If housing is too expensive for most people, is the government supposed to do anything?

That depends on who runs the government. If its the kind of government that wants the rich and powerful to be richer and more powerful, then Hello No! Let the poor b@$t@rd$ freeze to death in the dark. They shouldnt have been born poor in the first place.
Otherwise, Yes. Public housing and such support programs end up helping the whole economy and enlarging it an more skilled people are created, encouraging industry and the whole thing feeds on itself. But you have to think long term for that..😷