Time Machine backup keeps restarting over and over again
This problem appears to have been caused by either lack of free space, Spotlight, or both.

First I had to free up space so that I had at least 30GB free. Then boot in to recovery mode (+R on start), open the Terminal and disable SIP (System Integrity Protection) through this command:

csrutil disable


Then I disabled Spotlight through this command:

sudo launchctl unload -w /System/Library/LaunchDaemons/com.apple.metadata.mds.plist


After doing this I was able to finish my backup.