Losing a server does not lose the backups it sent. The copies on your buddy's server are kept, and a replacement server on the same HexOS account can pull them all back with one action. You do not need any access to the old machine.
When a server that sent backups is removed from your account, its copies on the destination stay and are shown as Retained backups, with the note "These copies belonged to a server that was removed. They stay restorable until you delete them". The connection card reads "Backups retained from a removed server".
Ownership is proved by your HexOS account. As long as you can sign in, you can recover.
Danger: Recovery depends on your folder passphrases. Encrypted folders in a retained backup cannot be opened without them, and there is no reset. See Folder encryption.

HexOS moves the backup to the new server and starts restoring every folder in it. Progress shows in the activity center. Encrypted folders ask for their passphrases before they finish, the same way a single restore does. See Restore a folder.
Warning: Transfer restores every folder in the backup. Make sure the destination pool has room for all of it.
Info: If the server holding the retained backup is offline, the dialog says so. Wait until it is online and try again.
Transfer is on your own account, because you own the data. Your buddy sees the retained backup on their side with Remove as their only action. Ask them not to remove it while you recover.
If the destination server is the one that is gone, the offsite copy is gone with it. Your own files on your own server are unaffected. Set up a new backup to another destination as soon as you can.
Help: Recovering and something is not behaving? See Backup troubleshooting or ask in the HexOS Discord Community.
Contribute: to help to improve HexOS documentation join the #Docs channel on Discord today! Send feedback, suggestions or contribute a guide.