I'm looking to buy a DS216+II to store family photos and videos, movies and music, important documents and the like. Part of these (especially the family stuff and the documents) I'd like to backup to another NAS, just in case one of them breaks down. Can this be done? If so, how do I install a NAS as a backup of another NAS running DSM6?
It is a standard feature in DSM.
You need the "Hyper Backup Vault" package on 1 of the Synology devices to act as the backup target.
On the other your run "Hyper Backup" to make the backups.
You can specify exactly what part of the NAS you want to backup and how often.
Alternatively (if you are comfortable with that) you can simply enable the rsync service on the target and do command-line backups with rsync.
In that regard DSM is just another Linux distribution.
Another option if you don't want to buy 2 Synology devices: If you have cloud-storage somewhere you can backup to that. For most cloud-storage system there is a sync app for DSM available.
Personally (3 Synologies at home) I combine both techniques: I backup the critical folders (mainly photos) of 2 of them to the 3rd Syno (using Hyper Backup) and from there I send an extra copy to OneDrive with the OneDrive cloud-sync app.
I have a 1817+ backing up to a 1815+ using "Shared folder Sync"
Scheduled 2x a week at the same time the 1815+ is scheduled to turn on.
It then turns off several hours later but once the first backup is done it takes less than an hour.
The folders are identical on both NAS so if #1 packs up then #2 has all the same data in the same folder layout!