I lived in Falls Church for 2 months when I first moved here (temporary housing that my company gave me). Now I'm in Fairfax.
Falls Church is an ok town. Metro area - very city-like. You'll have a bit of a drive to any good boarding facilities. Towards the west, you'll be looking at probably Fairfax Station, Manassas, Aldie, or Leesburg. I board in the Aldie area.
I think there are some down towards the southeast that might be closer to you depending on exactly where you are in Falls Church, across the bridge in Maryland, but I'm not very familiar with that area.
There might be some south in the Woodbridge area, too.
Good luck with the search.