The Big Wheel Bikes in DC is never open. You can rent a bike from the Arlington location (it's at US-29 around Spout Run Pkwy). But the pickings are slim, and the rentals are kind of expensive. I tried renting a road bike for a friend who was visiting, and the bike was too big for him and a POS. It was a pain for me to try to tune it up better for him and to fit it to him. Then his ride was pretty uncomfortable. You might do better with a bike tour bike from downtown and just tooling around instead of trying to get in a good workout without your own bike.

There are plenty of shops w/in the District.
Capitol Hill Bikes (at Capitol Hill / Eastern Market)
The Bike Rack (Logan Circle)
Revolution Cycles (Georgetown)
Bicycle Pro Shop (Georgetown)
City Bikes (Adams Morgan and in MD in Chevy Chase)

and there are a few others in the nearby suburbs