Many 1.25G eMac were plagued with a near-fatal logic board defect. Look at your serial number (inside the optical drive door) and see if it falls within these ranges:
- G8412xxxxxx- G8520xxxxxx
- YM412xxxxxx - YM520xxxxxx
- VM440xxxxxx - VM516xxxxxx
If it does, you likely have the curse. Apple maintained a repair extension program for this issue but it expired over two years ago. If this is the issue, there is basically no practical way to fix it today without spending the cost of 2 to 3 used eMacs.
NOTE: if you Mac is a refurb, the serial number will not look like the about formats (starte with R or RM instead) and the list doesn't help.
Sometimes you can see the problem through the RAM door on the bottom of the computer. If you see capacitors with brown or black goop on top as in those images:

[click image to enlarge if needed]
then you are pretty much out of luck, unless you live within driving distance of Eugene, Oregon. A place there repairs rather then replace the logic board for about US$130 if you don't have to ship, which could easily add US$100 to the equation.
Even if the caps under the RAM door are unaffected, there may be ones you can't see that are. It's a tough problem.