The problem seems to be common. Time Machine won't start: i.e. "it"... or the "Stars Wars" thing...
The following process eliminated the "error -6584"
Brand new out of the box, 2TB Time Capsule 4th Gen, backed up my iMac 24/4GB/10.7.2 over ethernet with no problems or errors. Time Machine did several incementals over night while I slept. When I returned from work, ready to reimage the OS, I discovered TC hung on an hourly incremental. When I tried to re-establish connection, no dice (error -6584) .
Several attempts to open the volume failed. I could manage all the preferences and configure the Time Capsule. I just couldn't open the back up or start Time Machine (app).
I dropped the %HOSTNAME file from the archive folder in Finder into the Disk Utilities image window to verify and repair, but received "file in use".
Restarting ("reset") the TC oddly resulted in renaming the volume label to "Unamed". I open Airport Utilities and renamed the archive folder to "Data" as before...
Now able to mount (login into) the volume... But it shows "--" in available volume space and no "Oldest" or "Latest" backups are displayed... Not a good sign. Waiting for backup to start..... Looking for backup....
Odd, instead of failing now, it started an incremental. Done!
I am now able to open Time Machine (or that cool Star Wars thingy that takes you off somewhere) and review all the backups starting with the first one I created before I went to bed last night.
The question seems to be what created the problem. I suspect my iMac went to sleep (shut down HDD) in the middle of an incremental. That may have set the "in use" flag on the archive and OS X didn't know how to handle that when I woke the machine up.
I hope this helps, and corrects my problem permanently.