Rats; a defective power outlet or surge protector would be easy to replace; a defective eMac power supply is not. Or more accurately, it can be replaced, but it's questionable whether it's worth doing so. It could also be something other than the power supply proper, including a dead PMU (which is why Apple took the PMU reset step out of routine PRAM battery replacement --- pressing the reset button too long or too many times can kill the computer entirely).
Given that there were no advance symptoms of any problems, that the computer had been left in sleep mode and was found shut down, and that there's no fan or anything else on attempted start-up, something in the power supply supply chain from wall outlet to the eMac's guts is indicated. Could there have been an unnoticed power surge the night the eMac failed? Common low-end surge protectors actually lose effectiveness over time, and they can pass damaging power spikes through while continuing to show a green status LED. The best case would be if a power spike hit the sleeping eMac and blew out the internal fuse that, if I recall correctly, is inside the power supply. A replacement fuse is cheap (the labor cost of getting at it and testing if that's indeed the cause of the problem, however, isn't).
Complete and inexplicable power failure such as yours is pretty rare, not that that's any comfort (I know firsthand what it feels like to discover that your kids computer is suddenly completely unresponsive --- and on Christmas morning, too). You may want to look over the first item under "Assorted Links" at
Apple eMac Upgrade Guide, which together with that link will give you an idea of what's involved taking an eMac apart. You can check with any local Apple Store or
Apple Authorized Service Provider (AASP) to ask about getting an estimate, but unless it's just an internal power supply fuse, it's almost certain it will cost more to fix than it's worth. Taking the hard drive out of the eMac will at least let you put it in an external enclosure so the data and applications can be used with a replacement computer.
Whether you can get the eMac back or end up with any kind of replacement, one thing you can do is pick up an Uninterruptible Power Supply (UPS). Compared to common consumer outlet strips, that will serve as both a better quality surge protector and as protection against power dropouts and brownouts. Even a low-end UPS provides that protection. Higher end ones give you more run time on battery, but even the low-end ones can be connected to the computer and Energy Saver set to gracefully shut down the Mac is power is out for more than a minute. A basic
APC BackUPS 350 UPS can be bought for around US$40 on sale, which is cheaper than a good quality surge protector (e.g.,
Tripp Lite Isobar line).