Depends on your definition of classic. Sure, $250 won't (usually) get you a Colnago or De Rosa, but if you are patient and informed, it will eventually get you a nice vintage bike.

I have bought a Peugeot PSVN, a Peugeot PX10, a Cannondale R1000/Mavic, a Motobecane Grand Record/Shimano Arabesque, etc. each for under your $250 goal. I live in an expensive city (as in, bikes are priced high here), don't go to thrift shops/garage sales, and am not the most disciplined of buyers. I'd never survive as a bike flipper, I wager I am one of the worst at finding deals. Yet there are some decent bikes in my garage for the cost of a new Surly Cross Check. Admittedly, some of these bikes also needed work but they were all complete and some were just grease, cables, and maybe a pair of tires away from being perfectly good riders.

Just read and read, here and on Sheldon Brown's fantastic site and on randyjawa's site. It may take awhile to figure out what you want. Know what? The figuring out is the fun part!