Pretty much all of Arlington is within a reasonable biking distance of the Pentagon.
Check out:
http://www.bikearlington.com/
and the WABA website for good bike path maps.
The Mt Vernon Trail goes pretty close by the Pentagon, and links up to all the East-West trails that run into DC (Custis, 4 Mile Run, W&OD, etc)
Affordable in Arlington is a bit of an oxymoron, and I don't know your budget, but here's a short listing of the neighborhoods I know well in order of where I'd prefer to live, if I had the choice:
1) Clarendon- very pricey, starting to get a bit too crowded, but very walkable. you can live on a quiet side street and be 5 min walk to a handful of decent restaurants, bars etc.Custis trail nearby
2) Courthouse- similar to Clarendon but less yuppie, Custis trail nearby
3) Ballston- more corporate, but located in a good spot, Custis trail nearby
4) Westover/East Falls- where I live now. less places to walk to, located on the metro line (as are the ones above) and on 66 and the W&OD trail
5) Columbia Pike- a little cheaper, a little more 'sprawl' but lots of condo conversions going up