Sure, you could get a piece of land in Siberian tundra at any time, I would not call that housing.
Moving to a city was way more complicated than in capitalist US. You could not simply buy an apartment. You had to be allocated an apartment by the government. And you needed connections for that. Or bribes. Ideally both. If you think your local rabid Republicans do not care for little wage slave men, you never experienced USSR, it was like that but 100x worse.