Some time ago, my old Time Capsule died. It simply did not start. I read that this is an issue that plague all older (first and second generation) TCs. The culprits are four electrolytic capacitors inside the switching power supply, two 1,500 µF, one 1,000 µF and one 470 µF. The 1,500 µF were badly swolen. I finally repaired it today after I ordered the spare parts from Mouser:
Unfortunately the plastic wrapper simply dissintegrated while I was trying to open up the power source. It is unbelievable such a cheap solution was chosen by Apple to package the power source. I had to cover the entire module in electrical tape, not before I completely wrapped the source in kapton.