If I were trying to see how cheaply I could build a cross bike, I'd be scanning craigslist for an early to mid 90's mountain bike, before they started having suspension forks and V-brakes. I picked this up for my daughter for $40 last fall:
It came with cantilever brakes and I think the frame is pretty decent. I couldn't talk her into drop bars and the controls it had were shot, so I had to switch to V-brakes to use the stuff I had on hand, but I think I could have built this into a good 'cross bike on a very tight budget. A lot of these older mountain bikes even have flatter top tubes, but this was a 13.5" version. Like look at this one that I found on Google:
Add skinnier tires and drop bars if you want 'em and that's a cyclocross bike.