Time machine can't find backup, drive looks blank, First Aid Fails, can't unmount

To unpack the summary in the title a bit.

My Time Machine backup has started failing on a fairly new hard drive.

Time Machine reports the following soon after it tries to start a backup:

"Backup Not Completed The backup could not be found on {drive name}"

When I look at the drive contents in the Finder, it appears completely empty:

If I try to run Disk Utility First Aid on the drive, I get the following failure messages:

"First Aid process has failed. If possible back up the data on this volume. Click Done to continue.

"POSIX reports: The operation couldn't be completed. Input/output error. : (5)

"Operation failed..."

If I try to run First Aid on the sole partition, I get the following failure messages:

"First aid could not unmount on of the other volumes in the volume's container. Click Done to continue.

"Volume could not be unmounted.

"Unable to unmount volume for repair. : (-69673)

"Operation failed..."

This leaves me without my backups data, without the ability to back up, without the ability to eject the drive safely, and without the ability to reformat the drive if it should come to that.