OK, I wanted you to try the easy fix first. There may be a much more ominous problem at play.
Without posting your serial number, see if it falls within these ranges:
- G8412xxxxxx- G8520xxxxxx
- YM412xxxxxx - YM520xxxxxx
- VM440xxxxxx - VM516xxxxxx
A large number of eMacs with USB 2.0 and 1.25Ghz processors (and a few non-cataloged 1.0Ghz units) have a near-fatal logic board flaw that can produce symptoms similar to yours. There was once an Apple Repair Extension Program to fix this (new logic board) at no cost but it expired in 2009 as I recall.
The specific problem was bad (leaking) capacitors on the logic board. Some affected capacitors may be evident when viewed through the computer's RAM door on the case bottom. They look like this:
[click to enlarge image if needed]
This affected other computer brands as well as Apple and is not user-repairable unless you have great skill at desoldering small parts from logic boards and working around dangerous, high-voltage CRT circuitry.
Today, the cost of repair is several times more than any used eMac is worth,assuming yo ucan find a pulled, unaffected logic board.
I hope this is not at play with your computer but I fear the worst. The serial number check is usually the best indicator, as Apple knows those computers were affected.
PS: the serial number is on a sticker inside the optical drive door. The about list will not help if the serial number starts with "R" or "RM"--these are used on Apple-refurbished computers.