You can even start with an old laptop and external drives. Plenty of people shuck them anyway, so you're not exactly overpaying. They'll just be a bit slow. But if you're mostly planning on streaming video that doesn't matter too much.

If you do have a little to spend, you can do a 'naskiller' build. Just search for it and pick one. Basically some people put together lists of cheap, reliable used hardware you can get to build a pretty great nas, with different flavors. 'Quiet' 'fast' 'compact'. I built one and went all out. I think it was about 600 bucks without the drives. Build up from there.