I'm a little reluctant to pimp products as solutions to our problems (NOT a mustachian way to do things, it feels) but I do know you can find these on Craigslist from time to time. Use your judgment.
A few years ago, I bought one of these:
http://www.amazon.com/Haier-HLP21N-Pulsator-1-Cubic-Foot-Portable/dp/B002UYSHMM/ref=sr_1_1?ie=UTF8&qid=1343018683&sr=8-1&keywords=haier+washerIt's a small washer that takes up about 18" x 18" of floorspace. Someone creative could build a little enclosure or top and make it usable flat space in a kitchen. I kept it in a corner of the kitchen when I lived downtown, and now it sits in our relatively huge closet.
It'll fit almost the same amount as you can run in a standard washer, and it does a good job.
At my old apartment, we had washers in the basement. I was in a third-floor walkup with extra-tall ceilings on each floor, so lugging my laundry up and down from the basement was a pain. I have always preferred to hang my wet laundry instead of drying it, and lugging a couple baskets of wet clothes to the third floor was annoying.
To wash a load cost me $1.25 when I lived there. Where I live now, it's only $0.75 a load, but still more than I care to pay. I do maybe 5 mini-loads weekly, which includes my clothes, our towels, cloth napkins, dishrags, sheets (LOTS of sheets...Cal King bed). That's probably 4 normal loads in a normal washer.
At my old place, that would be $260/year. At my new place, that's $156/year. I have had my washer for about three years now, half there and half here, so in total I have saved approximately $540. When I bought the washer it was on sale for about $200, which means I have saved myself about $340 over three years.
It's cute, effective, and easy-to-use. Some friends of mine bought one and a matching apartment dryer after they saw mine and have LOVED them. You don't need special hookups - you just hook it all up to the sink. Very easy. And it's no more work than you're already doing, and perhaps less. Instead of going to a laundromat and wasting that time, you can putter around the house and get things done.
The only thing is that you have to be aware when you hook it up. Mr. PS, who is unbelievably ADHD and breaks a lot of shit, did not put the drain hose in the sink once and flooded our closet. The business beneath us and my landlady were both PISSED. So he's not allowed to use it anymore.
Do any of you apartment dwellers have a great laundry system?