Could get some cheap dumbells at some point as you said, store them under your bed for instance.
Get a door-frame gym...So you can do pull-ups / chin-ups / wide grip pull-ups which will work your back and lats.
Squats can be done with some dumbells, or hell, some bottles of water or a bag of sand
Again, with the door-gym things you can do push-ups and dips as well, which will work your tri's and chest if done correctly.
No it won't turn you in to Mr.Olympia, but it will get the ball rolling, and to be honest if you can get good at pull-ups for instance it's a good starting point, it's something I need to work on now in all fairness.
Good call on the pull up bar, my friend had one for a bit of time. Could crack 15 pull ups and around 20 chin ups. could manage 5 assisted one hand pull ups (gripping wrist with other hand).
Always had the fear of falling flat on my arse though!
This is my room for reference: