Follow the link above for lots of ideas. I like to take advantage of good, canned products like salmon and tuna, even chicken, but be sure to check the ingredients list. While they are not super-cheap, they are far less than most fresh options and are good without a stove or refrigerator.
Thought I'd answer from a bit of personal experience (I'm new on the board myself and haven't read all the archives.) I've got about 10 years eperience living on my own, and I've just recently gone paleo (about 5 months ago). I found out quickly that paleo is actually a less expensive way of eating - processed foods cost more in the long run. My advice is to make sure you eat what you buy. It's very easy to buy too much produce and then let it go to waste. Take the time to plan your meals for the week, and then shop to the plan. After a while you'll get a hang of it and won't have to spend so much time planning. I can now run through my farmers market and get everything I'll need for the week and risk little or nothing to waste.
The other option, if you're close enough to a fresh food source, is not to buy too much in advance. When I started, I would swing by the grocery store every morning to buy the produce I was going to eat that day. It gave me an appreciation for making good food choices. Not the best strategy for saving money right away, but it helps you learn some good lessons.
Make as much of your own food as you can. Don't buy yogurt, kefir, pickles, broth, etc when you can make them yourself for a fraction of the cost.
Buy meat and chicken with bones and skin and stretch them to cover several meals--roast one night, salad with meat a second night, broth or soup a third. One chicken feeds my family of 4 three meals, at least.
Buy vegetables in season and can, dehydrate, or freeze extra. Buy canned fish on sale.