If you don't mind parking on the street, many residential streets in DC are zoned for permit parking (residents of that neighborhood only). Your girlfriend may be able to get you a permit and you won't have to pay for a garage. Not 100% safe, of course, but I've never had a problem with vandalism or theft while parking overnight on DC streets.

If permit parking isn't available, there are some unrestricted on-street spots in various parts of Arlington near Metro stations. You should check on your car at least once a week to avoid being ticketed/towed, though, as the signs change every so often (i.e. recently, a long block near the Rosslyn metro was rezoned from unrestricted to 2-hour parking).

The best place to look for a street spot is off North Oak Street (take Lee Hwy. west from the Rosslyn Metro and turn right just past the Key Bridge Marriott). This includes the blocks that were rezoned for 2-hour parking but as of last week, there are still several side streets in this neighborhood that allow free, unlimited-time parking. Or you can try the streets south of the Court House metro, between Wilson and Arlington Blvd.

The biggest drawback of this plan is that storing a car long-term on an Arlington street may require you to purchase an Arlington County decal and pay property tax (which isn't much in VA, but still an extra expense and hassle). Arlington parking cops are anal about enforcing the decal requirement, even if you have out-of-state plates. This may change soon, however, as many Virginia localities are eliminating decals altogether. But don't be surprised if after you're parked in the same space for a while, you get a ticket for no county decal.

If you don't want to park on the street, there are many garages near GW, and many across the river in Arlington, but they're quite expensive compared to almost anywhere (other than NYC). Not worth it, in my opinion, unless you are very worried about the security of your car and/or don't want to get the bumpers dinged up.