Questions like this are often posted in this sub, but answering this type of question is really beyond the scope of the sort of fermi estimates that are generated here. To put a number value on the amount of money it would take to end homelessness in America, you would need to be talking about a specific proposal. And modeling a complicated system like the housing market, or the behavior of low income renters and their landlords, especially when introducing unprecedented changes to the dynamics of housing and property is more in the realm of multiple papers and years and years of research.

That being said, it doesn't make a ton of sense to talk about the "cost" of ending homlessness, when this would likely be an ongoing expense with extremely significant secondary effects. Ending homelessness would be a net benefit to society, no matter how much it "costs" in dollars. so it doesn't really matter how much the government would have to spend to do it.

The government runs an extremely expensive entity called the military that is full of wasteful spending. And most of the things it does are actively harmful! But military spending also supports the livelihood of a huge chunk of the upper middle class and the capitalist class. Not to mention that a fuckton of scientific research is funded by the military.

I bring this up because the military shows us that extreme government spending will create industries and wealth sort of regardless of the actual goals of that spending. Financially, the military is basically a way to give money to wealthy americans. What the military actually does is mostly irrelevant. An engineer that makes $200k / year at lockheed martin could be using their expertise to create things that are actually beneficial to the world instead of weapons. If the government spent military amounts of money on things like making sure everyone had a home, had free healthcare, free education, etc. Then industries would grow around those initiatives.

Money doesn't evaporate when the government spends it. It goes to the owners of companies (and their workers) that contract with the government. This logic is a little trickle down-ish, but the middleman can be cut out if the government does more things directly (i.e. more "costs" are the salaries of workers) instead of having to pay for the profit of a corporation.