Disk Utility 'Repair Disk' /'First Aid' check the drive Directory for internal consistency. If all errors can not be repaired, Mac OS considers the drive damaged and its Directory (and therefore its contents) are not to be trusted. In that case, the drive will not be mounted and is unusable in its current condition.
There are many different errors that could cause Directory corruption, and if the corruption is too complex, Disk Utility may not be able to repair the corruption. Disk Utility errs on the side of data integrity, so it will not make any "risky" repairs (i.e., ones that could drop files out of existence, or create new ones that may never be intended to exist.)
Software errors, power failure while writing, and Bad Blocks are common reasons for directory corruption to occur. Most of these can be eliminated by erasing and re-writing the drive. In the case of Bad Blocks (caused by magnetic regions going Bad) the drive controller can substitute spare blocks from its reserves for blocks found to be Bad, but only when the block is re-written with new data.
Under normal conditions, erasing your only Backup drive leaves you "working without a net" during the time to create a new backup, so that is not a good general recommendation. Some files on the backup drive might still be accessible with Data Rescue software, if that became necessary. Data Rescue software ignore that the drive is unmountable, and do low-level I/O to get the data, then hope it has some integrity.)
So the general recommendation would be to ADD an additional Backup drive (new or old, large or small) and not erase your old one until you have a known-good, Trusted Backup to take the place (at least for a while) of the one that is damaged. Time Machine from 10.8 onward can add an additional drive, create a new stand-alone backup on it, and then attempt to "alternate" drives on successive backups. Using this standard approach in this case would mean you are covered by a backup (however damaged) while you make your new new one, then can erase the old and hope the old drive still works.
If your Backup drive is not quite meeting your needs, now is a great time to introduce on that will last well into the future.
