When I was in graduate school, my weekly grocery budget was $35 a week, that included things like shampoo and toothpaste.
I bought rice and beans (the dry kind you soak, not cans) in bulk from a nearby Asian market. To save enough to do that, I only ate potatoes for a couple of weeks. I figured, if the Irish could survive like that... I saved enough to buy a 25 pound bag of rice and smaller bags of black beans, chickpeas, red and green lentils, and pinto beans. Those lasted me for a couple of months.
Then, I just bought eggs, canned tuna, oatmeal and seasonal produce each week. I only bought veggies and fruit that were in season and therefore cheap. Cabbage was often on the menu because it was both inexpensive and healthy. I got day old bread from a local baker - it wasn't much cheaper than the stuff at the supermarket but the quality was MUCH better - real ingredients instead of preservatives and crap.
I got rid of the car and either walked or took public transport and that saved me a LOT of money. I also did not have TV, internet or a phone. If I needed to make a phone call, I did it from my office at the university.
I was able to live on $650 a month by doing this. It wasn't easy but I made it work. It was a lot easier for me though because I knew it was just temporary, while I was in grad school. I think it's MUCH tougher if you don't have a light at the end of the tunnel, or if you are forced to do it vs. voluntarily going to grad school.