I suspect it is a national or semi-national chain, as there are also a couple of these in Birmingham, Alabama. On is at Five Points South, the other at The Summit. They serve nifty fountain drinks like cherry Coke (with real cherry syrup) or (my personal favorite), Vanilla Dr. Pepper. (Good milkshakes, too!)

The atmosphere is similar to Steak 'n Shake, if you happen to have one of those available for comparison (but unfortunately, not open 24 hours).

I don't know about the ones in the D.C. area, but at the ones down here, if anybody plays "My Kind of Town" by Frank Sinatra, the whole staff drops what they're doing for a moment to lip-sync along with the ketchup bottle. (Cute choreography, too!)