Off means "no power": http://support.apple.com/kb/HT4600
It is possible its power supply has failed. An Apple Store can make a definitive diagnosis, but without test equipment there is nothing more you can do.
Lots of people extract its internal disk drive and press it into service in another enclosure. Chances are it's still OK even if the power supply is not.
I was just scouring the other posts on this topic. Seems to be the case. I'll bring it to an Apple Store, but it is out of warranty. Trying to extract the disk to use in another enclosure seems to be a suitable idea. I really wanted it for my wifi base station, but my Airport Express is working just fine for that. I never liked using the Time Capsule as the backup disk in Time Machine. It was much too slow. I'd prefer to use a connected (Firewire 800) disk for Time Machine backups and restores.
You're right about the time it requires to back up over wireless - an external drive is a better idea. Buy an external enclosure for $40 or so and your Time Machine backups can continue as if nothing happened.
Here is one procedure for removing its hard disk: