If you log in through your Mastodon server's website and go to Preferences, there's an "Import & Export" section where you can download Follows (people you follow), Lists, Blocks, Mutes, Domain Blocks and Bookmarks. These can be backed up and re-uploaded to another account if needed.
You can't back up your followers (people who follow you), but you can transfer them from one account to another.
When you move accounts, you basically transfer followers and then upload backups.