>if you had a certain amount of cheap housing that even drug addicts and mentally ill people could afford.

That was absolutely free, $0?

>Like some housing that was built to be closer to "indestructible" than "nice" and was small, like 200 - 300 sq ft.

We already have those housing units. They have concrete "beds", toilet-sink combos made out of stainless steel, and a thick steel door that can't be burned by arson. Nothing flammable is allowed within them (save the non-homeless person). The technology exists, and the homeless are actually welcome there. People get mad when they are homed thus, though.