I think the most likely culprits are a failing hard drive, or the bad capacitors problem that struck the first models of G5 iMac and the G4 eMacs.
First the hard drive. If it's the stock drive, it's a prime candidate for failure (most drives last 3-5 years; think of them as consumables, like light bulbs.) Unfortunately the SMART status used by Disk Utility is next to worthless for the most common failure I see, bad blocks (sectors). If you go to versiontracker.com and download SMART Utility, it will let you launch a few times as a demo. SMART Utility reads the SMART attributes of a drive directly and will tell you if your drive is suffering from a failure or imminent hardware failure. Of course, you'll need to boot to an external drive for this. SMART attributes can only be read on an internal drive, so putting your emac in Firewire Target Mode and connecting it to another Mac will not work in this case.
Second, the caps. Your symptom is actually more indicative of the hard drive problem, but if you get your computer booted and it freezes, shuts off, or has video artifacts like lines across the screen, then it could be the caps. If you're comfy taking it apart (no mean feat with an eMac, the bitty power cable is a bear to plug back in) you can examine your caps for bulging and leaking.