The only reason you enjoy cheap food is because of the free market.
Housing in many countries is not part of a free market but part of a monopoly where the government decides how much, how many, the what and the when the houses can be built. If it was a real free market, the issue would have been long solved.
Housing should be a commodity. Commodities are fungible, plentiful, and easily tradable.
Housing right now is the opposite of a commodity: it's a boutique investment for individuals whose price is propped up by regulations that artificially limit supply.
Surely, if we are decommodifying basic needs, our first priority should be food, not housing, right? How has nationalization of agriculture worked in the past?
Who builds all those bridges, roads, power plants, and other public works projects? Maybe them?
Also government formed co-ops. Find people who want a flat and agree on rent before you build it. As the loan is paid off rent becomes substantially cheaper. The second best time to plant a tree is today.
It also doesn't need to be the entire housing stock, just enough to help keep rent low (see Vienna).
Governments directly, cooperatives or non-profit organizations. The Churches used to be pretty active in that space as well to invest tithes, with rents paying for common-welfare projects.
Rent extraction is utterly counterproductive for a society.
Housing should not be a commodity.