I was wondering if it's possible to run a backup game off of a networked computer's hard drive, like how XBMP can stream a video / audio file from a networked computer. Basically, I'm wondering if you need to upgrade the Xbox's main HD to play more than one backup at a time (I guess you could always store them somewhere else and FTP files back and forth).

No you can't stream games over a network.

Either upgrade your hard drive or keep the backups on your computer and transfer it every time you want to play one.

Seriously, the XBox stock drive is barely large enough for a sub-par pr0n collection. 80 Gig Maxtors or Seagates are super cheap at various retailers. Much less hassle just to swap out the drive than figure out a method of streaming games over the LAN.