You can run your pup off-leash at Congressional Cemetery, about 25 acres. There is a fee of $10 if the caretaker sees you and doesn't recognize you as a regular. You can also sign-up to be a 'member', but there is a not so surprising huge waiting list.  

I've taken my dog there several times, and its a nice walk around and completely fenced.

Battery Park near Foxhall Road

There is one on Calvert St. Not very grassy though so they might get dusty.  

Map of DC area dog parks + several review sites.