It can be done. I have a decent HT in 13x11 room. Bigger is nicer and my new house will have 17x12 space but small room is better than no room at all and can fit a projector. Your surround sound is not going to be the best possible but you won't have to invest a lot to get great bass.
I have my projector (Benq 7800) in a one bedroom apartment. The area the projector is in measures about 13x12. It has worked out fine for me.
One suggestion I would have though is to pay close attention to the noise level of whatever unit you get. In a small space, chances are you will be close to it. On my Benq when the fan kicks into high gear (maybe 20% of the time) the noise can be a bit distracting.
First off, with your budget you'll need to buy a 480p projector. The recommended seating distance with a 480p projector is about 2X the screen width. That gives you a screen width of about 5'(60") minimum. You could probably go a little bigger, but you won't know for sure until you check it out.
Even so, you might want to consider how big you really want it - many films can't survive the Imax Experience(TM).
But yes, there is no reason why you can't have a projector in a small space. Because of certain issues, I sit 8' from my screen, and it's approximately 72" diagonal (~1.66:1 screen; again, other issues.) On the other hand, it is acoustically transparent with the center speaker behind the screen's center.
Sami's right, there are a couple LCD 720p projectors available for $1200 or less, but I wouldn't buy one. One is the OLD Sanyo Z2, and the Infocus 5000.
Try to get a 720p DLP projector but if you can't find one in your pricerange I would rather get a 720p LCD than a 480p DLP. HD discs are going to be readily available by next year and they look simply awesome, even on a SP5000 (which is not that great of a projector).