In order to live on a small amount of money, it's important to think outside the box. If you try to rent an apartment and buy all of your groceries at a store, you'll never be able to make ends meet anywhere in the U.S. without public assistance, but if you're willing to live unconventionally, you could probably get by on <$250/month just about anywhere in the world.
Volunteering on an organic farm through the
WWOOF Program would make your two biggest expenses, food and accommodation, free. It's possible to "WWOOF" on one farm for an entire season or hop around from farm to farm, only staying on each one for a few days, weeks or months at a time. Probably you could afford to live anywhere in the world on $250/month if you were willing to help farmers out in exchange for a place to stay and delicious, healthy food to eat...
There are all different ways I could think of to survive in the U.S. on <$250/month, but they all involve some sort of non-conventional living arrangements. For example, my wife and I once lived on the Island of Hawaii for 6 months and only spent ~$250 each/month, and we had an amazing lifestyle. We did a work trade for our housing. In exchange for free rent on our cabin, my wife and I each volunteered to work a 1/2 day per week helping the landowner take care of her property. We pulled weeds, planted fruit trees, fixed fences, etc., instead of paying rent. The property we lived on was, literally, covered with food-producing plants: bananas, papayas, avocados, guavas, vegetable gardens, etc., so we were able to glean a large part of our food from the land for free.
When my brother first moved to Gainesville, FL to go to grad school, he couldn't move into his apartment until a month after he arrived, so rather than stay in a motel/hotel, he rented a small storage locker to keep his valuables, and slept in his tent in various secluded spots near the university campus. In the daytime, he'd store his tent and other valuables in the storage locker, change clothes and walk around campus like a "regular person." Before dark, he'd head back out on his bicycle with his tent and camping gear in tow, look for a safe place to camp, set everything up quickly, just as it was getting dark, and go to sleep. Rinse and repeat. I'm pretty sure he paid <$100/month for the storage locker, and knowing my brother, he didn't spend more than $150/month on food.
Volunteering to help take care of somebody's house, farm or estate while they're away, possibly through an organization like
Trusted Housesitters would be another way to keep expenses low and possibly live on <$250/month. Again, any way that you can avoid having to pay rent makes it possible to get by on very little money.