Hi, Rachel
There are a few medium scale apartment complexes in Fairbanks (Wedgewood, Jillian Square, Sophie Station to name a few) but most apartments are in smaller buildings that are not heavily advertised. If you're hoping to find something near Eielson you'll be looking at an apartment in a duplex or triplex or a single family home. Most of the condo's in Fairbanks are predominately privately owned and not used as rentals. Honestly, your best best is to watch the classifieds in the News Miner (newsminer.com) or Craigslist (Anchorage.craigslist.org - covers all of Alaska). If you don't have any luck there let me know and I'll be happy to refer you to some property managers in Fairbanks that might be able to help.

Rachel, the housing management flight at Eielson recommends ahrn for rentals. The housing office can give you some good assistance DSN 317-377-2727 or Commercial (907) 377-2727.