I use Crashplan - it doesn't need to be on all the time, and your neighbours computer doesn't need to be on all the time (the one that has your USB disk plugged into it).
Indeed! You have two options. Either via network or physical disk. If you do it via network, Crashplan will perform the backup when both PC's are online. If you need to restore you can copy the backup repository from your neighbor's computer onto a physical disk and restore from that at home.
Alternatively you can simply use a couple of USB disks, set up as two separate destinations for the backup set in Crashplan. Keep one at home and one at your neighbor. Once a week or whatever you swap them. Crashplan will automatically detect the disk when you plug it in and start syncing the backup.
The best part of this is that the data is encrypted in either case, and IIRC you can do all this using a free account. The paid options only matters if you also want to store the data in "the cloud".