You can, you could have every computer in your house that connects to your network send it's back up to your external drive.
I know it's possible, I can't tell you how to do it, cause networking ain't my strong suit, but these guys here should be able to. I know this is how my pop-in-law does his. His desktop, 2 laptops, and a desktop on his living room TV all backup to the same external drive connected to his main desktop. and he can access that drive from any one of the computers. I guess it's basically making it a server of sorts.
it is not hard to setup.. you can have the the drive hooked to one PC, create a share to a folder of you choice, map the drive to the folder shared and it is like a local drive. Run backups.. and ur done. Instead of going to the external you can have one PC backup to the other PC so the external is not plugged in all the time (bad idea) then just have the other PC's backup include the other PC's.
I can help you setup if you want. If you ask Pinglo how to he starts talking about clouds. His head is always in the clouds. hahahahaha
Ideally, with an external drive, if you can get one that plugs into your router, then no need to have a PC on to share the drive as the external would be available at all times through the router. But what hxtr said will work as well.