Paul Kirtley wrote:
I might mention, this all occurred after I ran a "Repair Disk Permissions" of my Macintosh HD. The process encountered no errors, but repaired a slew of Disk Permissions in the process. When Done, I was no longer able to access my Time Capsule backups.
That's interesting -- something had clearly gone wrong on your internal HD to mess up a lot of permissions. 😟
(You might want to also reset the permissions on your home folder (Disk Utility only fixes permissions on system files and files installed by the OSX installer), per Resetting Password and/or User Permissions).
On the next backup, every file that had it's permissions repaired was treated as changed, and had to be backed-up again (Time Machine backs up the metadata, including permissions, as well as the data files). What then went wrong is anybody's guess, but something dire happened, or you wouldn't have gotten that message.
Does this shed any new light on my case, or am I faced with only one option...erase the backup disk?
Most likely, you've had a network or power problem, per the link Bob supplied.
There is one other option -- "archive" the damaged sparse bundle to a USB drive connected to the TC, per the green box in #Q6 of Using Time Machine with a Time Capsule. You can probably view and restore from them, via the Browse... option, per Time Machine - Frequently Asked Question #17, but remember, they're damaged, so if you happen to hit a corrupted spot, anything may happen.
Then erase the TC's disk via Airport Utility, per the green box in #Q5 in Using Time Machine with a Time Capsule, (but in this case, you might want to select the Zero-Out Data option. That will take longer than a quick erase, but if it fails, you'll know there's a problem with the TC's disk. If it succeeds, the disk may be ok.