Fix Android stuck on boot animation without deleting data
Stubbornness sometimes pays. Here is what solved the issue for me (and since I did not see this idea anywhere, I'm posting it here): it turns out that the (attempted) backup filled up the internal memory so that there was not enough left to complete the boot process. So I used the file explorer in TWRP 3.0 (Not sure if CWM has one, if not, just flash TWRP) to delete the backup and voilà, the phone boots again.

Obviously, you could also delete something else (make sure you know what you're doing), the point is to free up space on the phone's internal memory. In fact, I actually tried to move the backup to the external SD card but there seems to be a bug that (sometimes) lets the move command fail so what I did was copy and delete instead.