As Brian from Texas wrote:
"My TC did the exact same thing two days ago. It is 18 months old. After doing some research, I have found others with the same problem. At 17-18 months, the power supply goes out. I went to the Apple store today and they said too bad, out of warranty."
There are 6 pages of TC owners with similar issues, most of them happening after about a year and a half, so it looks like something is going on with the units.
This seems eerily similar to the 'dead airport express' syndrome of a couple of years ago where the products would simply fail to work and all the data and power lights went off. Sometimes there was an audible popping sound. Other times, the APX just died.
If you are having this issue it might be a good idea to report it to Apple and see if the weight of complaints gets them to look into this. Let us know too.
Thanks to Mike for the tip.
Update: As it's being discussed in the comments here, Lauren went ahead and investigated whether AppleCare for your Mac will cover an out-of-warranty Time Capsule. The answer may surprise (and please) you.