I don't know that area real well, but I don't think that there are any gastropubs nearby. In lieu of answers or while you are waiting you can search this board for Adams Morgan which you are nearby where there are tons of restaurants and bars. Perhaps you will find something.
If you do want to travel a bit, you can go to Granville Moores. It is a ways away, but is one of DC's few gastropubs. Teddy something (the owner) has been on Food network and he beat Bobby Flay in a Moule et frite throwdown.
Are you comfortable walking a bit? Go east down Calvert into Adams-Morgan and take a right onto 18th Street to Black Squirrel. Caveat is that I haven't been in about a year, but it's got a nice atmosphere and I highly recommend the duck spring rolls: http://www.blacksquirreldc.com/menu.html
i don't know if you still need this info, and if you are open to food beyond gastropubs ... but there are two good lebanese food options in the area (i used to live across the street from the omni). there is lebanese taverna and then there is Mama Ayesha's (which is on Calvert St. towards Adams Morgan).
Open City across the street from the hotel is a really nice neighborhood hang out place with decent food and free wifi.
In Cleveland Park (one stop on the red line) you have Palena a upscale restaurant which has a bar area where you can have ala carte and their burger and hot dog are really good. Dino is a good Italian restaurant there and also a couple of good irish pubs
2331 Calvert St NW, Washington, DC 20008