Sadly, I have researched my cheap lodging options in the DC metro should I ever lose my job and place too....I have come up with these (most will require you to have a car as public transportation is non-existent or very spotty)....and these are for a max of 2 people...if you have a family to support....Godspeed....
for 1500/mo or 400/wk or less (basically paying rent but the place will be much nicer than a motel...you can cook, etc) -
http://www.valueplace.com/extended-stay-hotels/locations/virginia/Gainesville-VA/
http://washingtondc.craigslist.org/nva/sub/1825564460.html
http://www.innsofvirginia.com/
(inns of Virginia in Woodbridge- ad says $315/week)
http://www.suburbanhotels.com/hotel-sterling-virginia-VA486?sid=Ks9u._Gi_tgIekg.1&sarea=&sname=&sstate=&scountry=&sradius=&slat=&slon=&schain=&exp=&scity=&sort=&type=&map=&srp=&amenity=&sstate_country=&city_code=&nadult=2&nchild=0 -surburban extended stay dulles- rates online are high but you can call them and talk them down :)
Now if you just need a roof over your head and need a cheap hotel just to sleep in....breezeway will work. Just keep in mind you can't stay more than 30 days straight, they will make you check out (albeit for a night) to reset the clock.
I can also (sadly) recommend (barely) any of these places that show up on this google search under "nearby places you might like" near/next to VA lodge in Alexandria, VA off route 1 just past the 495 exit headed south. you can get a single room for ~55/night and can probably work out something cheaper with management.
http://maps.google.com/maps/place?hl=en&um=1&ie=UTF-8&q=virginia+lodge+alexandria+va&fb=1&gl=us&hq=virginia+lodge&hnear=Alexandria,+VA&cid=2132146653503586579&dtab=2&ei=fFAxTP6dJYP78Ab7u_HLCw&sa=X&oi=local_result&ct=result&resnum=1&ved=0CCoQqgUwAA
granted, no wifi or not even a really modern room but it is a roof over the head with a crappy fridge/micro to just have a cheap place to sleep until you can find something nicer....
I have not looked further south in VA (don't know your transportation situation) but I am sure you can find a decent days inn, motel 6, etc to stay in for ~60 a night if you are able/willing to drive a bit. I would, frankly.
Good luck in whatever you end up doing and God Bless.