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
Affordable is the key factor. The Columbia Pike area has some affordable options and is a straight shot to the Pentagon. Just across I-395 and Glebe Rd. (not sure what it is called) are a good number of apartment developments that are just about two miles from the Pentagon and 3 at most from Crystal City and the Four Mile Run Trail.
The Rossly-Ballston corridor is a few mile stretch of condos, restaurants, offices, etc (urban development) that stretches from the edge of the Potomac River (DC and Georgetown are a short drive or walk over a bridge) at Rosslyn to the Ballston section heading west. Five metro stations line this corridor (Rosslyn, Court House, Clarendon, Virginia Square, and Ballston--my 'hood). Although the rents can be high in this area, there are a lot of good finds out there. Often, those buildings might not be brand new, are a several blocks away from the metro entrance and restaurants, etc. But really, what is several blocks. Take some time and look around. Don't be immediately discouraged by the high rents in some of the more modern and bustling areas. Keep looking, be open to the smaller and older buildings. You can easily save hundreds and hundreds a month if you don't care about granite counters, stainless appliances, etc.
Arlington is small. If there isn't a metro nearby, there is likely a bus route to take you to the trains, DC, pentagon, etc. Good luck.