Some eMacs had serious, near-fatal logic board issues tha could produce the grief you describe. Before we go much farther, let's see if you have an eMac in the affected ranges.
First, look for the serial number. This article shows where to find it:
The bad logic boards affect many 1.25Ghz eMacs and the seldom-seen "edu-only" 1.0 ghz version with the later USB 2.0 logic board of the 1.25g. The computers affected were supposed to fall in these ranges:
- G8412xxxxxx- G8520xxxxxx
- YM412xxxxxx - YM520xxxxxx
- VM440xxxxxx - VM516xxxxxx
If your serial number starts with "R" or "RM," then yours is a factory refurb and the above list won't help.
For security reasons, Apple doesn't want people posting there serial numbers here. Instead, simply tell us if your eMac falls in on of the affected number ranges. Then we can go from there.