I am a poor college student, and my budget is $20 a week at the grocery store, with one meal I eat out. I don't starve! I swear! Even then, people don't understand how I keep it so cheap. I just invite them to go with me and buy exactly what I do, but no one takes me up!
(Midwestern grocery prices, though.)
I avoid buying snacks/processed stuff other than crackers and cereal (which must be on sale), and mainly stick with just bread/produce(on sale mostly)/milk/eggs/occasionally (on sale) meat to be frozen. I stock up at Natural Grocer's in the city occasionally on things like organic chicken bouillon and at an Asian supermarket for noodles, soy sauce, sriracha, tofu, etc. Everything is amazingly cheap at those places, a lot of it is the same stuff you can get at the regular grocery stores, but priced for people that eat it all the time and don't consider it "specialty."
It would cost more if I went organic, but that is neither within my means or available for most items in my area. During the summer the farmer's market is open, and I save even more money! I also benefit from free veggies from my mom's garden. So, winter is more of a drag.