For folks living on the street, you absolutely cannot earn enough from a minimum wage to house yourself. At best, you could barely get by sharing housing with a few folks. Assuming you have addiction issues that lead to anti-social behavior, you will have difficulty maintaining relationships in share living spaces and maintaining your employment.
I think we would have a dramatically different situation with our homeless, if someone could reasonably afford a small one-bedroom or studio apartment at 25%-40% of the prevailing minimum wage. As it stands, in Los Angeles, median one-bedroom seems to be about $2,362/month, and a minimum wage job there, before taxes only pays $2,462 per month.
Thus, a minimum wage job, even if the person magically paid nothing in taxes cannot even come close to covering the median unit, by a factor of 2-4. In addition, that median unit likely requires further assets like a vehicle in order for a person to be able to access employment.
Because of this American development pattern, it is nearly impossible for people to even begin to earn enough money to maintain any sense of private living space.