I have lived in an apartment without washer & dryer hook ups or a laundry room since September 2010. I decided that when I moved in I would hook my washing machine up to my kitchen sink and air dry my diapers. I have been doing this for the past 9 months and I am happy to report that I haven't had any problems! At first I was worried about water pressure and the amount of time air drying takes, but both of these things are non issues.
I have been using different kinds of soaps. First I was using Rockin' Green, which I absolutely love, but once that ran out I didn't have the money to get anymore. Being a single mom means I am on a serious budget. Luckily I had a large box of Country Save, so I used that for quite a while. I recently joined this awesome site called Heartsy, I had some credits from referring friends so I was able to purchase a deal from Life Soap Simple and I got some natural citrus laundry soap for free (and a stain stick too)! I have been using this on my diapers and it's working wonderfully.
For my wash cycle I turn my cold water on and do a cold rinse. Once that's finished I turn my sink to hot and put it on the longest wash with an extra rinse, then for good measure I do one more quick cold rinse and then hang the diapers up to dry.
One good thing about living in Texas is the fact that it's been warm/hot for months now, so my diapers dry super fast when I hang them outside. During the winter months it took a little longer since I usually just hung them indoors, but I have never been without clean cloth diapers to use.
Even if you can't hook a washing machine up to your sink, there are ways to wash diapers if you live in an apartment. Don't let apartment living sway you away from using cloth. Depending on your situation there are diapering options that would be more apartment friendly than others. Do you live in an apartment and cloth? Share your routines!