My process for backing up the various desktop and laptop computers in my home is clunky. I currently just use an external HDD that I connect to each computer periodically and run the backup software that came with the HDD. I'm running out of space, so now's the time to make the switch to a better in-home solution (I don't want to use a cloud backup service).

Basically, I want some type of shared, expandable network storage that is accessible to all the computers in my home and that automatically backs itself up (preferably to another drive that I can remove and store in a separate physical location). I also want it to be simple and idiot proof; I don't want to spend a bunch of time setting it up and maintaining it.

Can anyone point me in the right direction? Bonus points for an Amazon link. Am I looking for a NAS? A RAID system?