If there is a trader joes near you they have a lot of items that don't need to be refrigerated, designed to bring to work for lunch. Also you can get away with eggs just being kept cool for about a week.

Fresh veggies, jerky, albacore tuna, silken tofu...

Cheese actually stores well at room temperature, and investing in a butter bell will allow you to safely keep your butter unrefrigerated.

Seriously? Buy fresh vegetables. Cook them as you would any other vegetable you store in your fridge, just don't buy as many. Vegetables can keep for quite a while out of the fridge, as well as eggs (which keep for a really long time on my kitchen counter). I lived without a fridge for over a year and there wasn't any noticeable change in my diet, only my lifestyle. Go shopping every other day, not once a week.

well other than meats and dairy, for the most part a lot of foods don't really need to be refrigerated unless you don't plan on eating them for awhile. You should probably be fine if you just buy less at the grocery store but go more often.

I also lived without a fridge for a year. Just cook for that meal. If you're having steaks for dinner, buy them that day, cook them and eat them. You'll have a lot less food waste. Veggies do last a while and so do pantry items, have a few vegetarian meals.

Peanut butter! ☺

Use a cooler and some ice for a few things and store the rest on the counter.

Eggs are natural tin cans, they actually don't need refrigerating, so you can have endless amounts of omelettes :)

Just shop more frequently and cook foods that won't give you leftovers. Buy produce and meat you need just for that meal.

Tuna pouches. Those things are so handy!

Pasta, potatoes, canned veggies. If you have a cooler, you could fill it with ice to keep some things cool or at least have ice for drinks. If you want meats, just buy them the day you want to eat them and cook them up right away.

I would ask around to see if any friends or relatives have a mini-fridge from college sitting in storage that you could borrow.

Awesome suggestions. I have notes to add on a few earlier comments:
* Eggs sold in Europe or straight from a farm keep better without refrigeration than eggs sold in US stores. US egg processing removes more of the shell's outer coating, making them more permeable.
* You don't need a butter bell to keep butter safely on the counter for a few weeks; any lidded butter dish will do fine without the bell's water hassle, and you don't even really need a lid. If you're really worried, use salted butter, which keeps longer. (A friend's family had a raw-milk dairy farm; they churned their own butter from it and kept a big block out, uncovered, on the counter at all times. It was great.)
* If you're desperate and can't borrow a mini-fridge, you can probably get one cheaply from Craigslist or your local classifieds, and given the circumstances can perhaps convince your landlord to cover the price. (Or, for that matter, to cover interim rental for a full-size fridge.)

Enjoy the adventure!

Ask your landlord for a temporary mini-fridge, I'm pretty sure it's in your lease that you need to have adequate appliances...

Get bread, toast a slice and mash avocado on it with a little salt and pepper. Add a slice of tomato if you'd like. Super delicious.

Eat lots of fruit and raw veggies. Apples, bananas, carrots, cucumbers, etc.

Slice zucchini ribbos with a vegetable peeler, toss in good olive oil and season with salt and pepper. YUM! Add any seasonings you like to intensify flavors.

You can make small batches of couscous which cooks up very quickly.

You may need to go to the grocery store every couple days since you don't have a fridge, but just enjoy the fresh produce! You might find many things you like.

You could also pick up some Lara bars for quick snacks. The Peanut Butter & Jelly bar, Blueberry Muffin Bar and Coconut Cream Pie are my favorites. Best part is, they are full of REAL ingredients. Nothing fake.

We just went through this!
We found out our fridge was toast by pouring a nice lumpy glass of sour milk! Not very pleasing to say the least.

Best bet is to store any dairy products in the freezer. Our freezer was hovering around 0.5 degrees Celsius so the milk was very cold but not frozen.
Find another freezer to store your frozen stock in.
Butter can be kept on a small plate with a bowl over top.
Buy meats and use them the same day. No leftovers allowed unless they fit in the freezer.

Also, put the freezer setting (if there is one in the freezer) on the coldest setting. This prevents the cool air from entering and cooling the fridge, thus wasting energy.

The cook in me would recommend only cooking what you can eat immediately and relying on grains and pastas and fruits and veggies so nothing goes to waste. The future lawyer in me, however, would check your lease and your state's landlord/tenant laws to see if you can coax your landlord into getting the new fridge sooner or purchasing one yourself and deducting it from your rent!

dry ice to keep the fridge cool?