Chosen Solution
I’ve been using this particular 3TB Time Capsule to host my wireless network as well as a good deal of content for access from different computers (date of purchase Early 2013 - Apple Store). Recently, it seems to randomly power off without warning. It doesn’t happen more than once a week, and afterward a simple power cycle gets everything up and running again. I was doing some research and it seems like older models have had problems with the power supply - but nothing seems to current. Any help would be greatly appreciated.
If you noted the problem was during or just after a large data move I suspect the unit is overheating. I would try cleaning out the dust build up using some can’ed air and a soft paint brush. Also see if the unit is getting enough ventilation as well.
Unfortunately, even the latest versions of the Apple base stations can still have overheating / power supply issues, that may not show up immediately. One option, if you have AppleCare in effect on a qualified product, the AirPort Extreme or Time Capsule will be covered for up to 3 years from its purchase date.