External HD shows "Invalid Extent Entry" in Disk Utility First Aid for the Time Machine "Backup" volume. Erase that Backup volume? External is a Seagate 2TB Backup Plus Desktop Drive for Mac. I'm running 10.11.6 on a Mac Mini. It's connected by FW800. Disk Utility First Aid for just that volume stalls after the "Invalid Extent Entry" note. 

I ran Disk Utility on the entire external drive (all volumes). Result: [All checks seem good, then...] "Problems were found with the partition map which might prevent booting. Operation successful."

Other 3 volumes on that Seagate seem to work fine. One of those volumes has secondary Time Machine backups. 

After restart, Mac without prompting says: "OS X can’t repair the disk “Backup.” You can still open or copy files on the disk, but you can’t save changes to files on the disk. Back up the disk and reformat it as soon as you can."

I assume I should erase that Backup volume but my concern is I'm afraid erasing that Backup volume might screw with the other volumes, and my only other TM backup, which is on another volume of the same external HD.
The invalid extent entry error basically means the directory on your drive is corrupted. In other words the index of what files are stored and where they're stored is all mucked up. 

This can be repaired but not without risk. You basically have two options if you still want to keep using this drive:


Use a third party disk utility that can repair the corrupted directory (e.g. DiskWarrior, Techtool Pro, etc)
Reformat the partition 


Note: I have no affiliation whatsoever with either DiskWarrior or Techtool Pro.

Regardless of the option you choose I would still recommend manually backing up any files you can still access to another drive. Obviously this is a must do if you opt for the second option.

As to your concern about accidentally reformatting the other partitions, this shouldn't be a problem as long as you're only selecting the particular partition (or volume) with the error.

More specifically:


launch Disk Utility 
select the partition in the sidebar 
click the erase tab
make sure the format in the drop-down menu is the one you want 
give it a name 
click on Erase