I'd like to figure this out if you can bear with me. I'm retired too so have time to research Mac stuff when I'm not taking a power nap!
There are two families of eMacs with two variants each; some have known issues. If you are up to it, please read this Apple article that lets you find your exact eMac variant by serial number:
http://support.apple.com/kb/HT2395
Without posting your serial number here, post what model the article says you have.
If the model returns as "eMac (USB 2.0)" that is a group with very bad logic board issues. Affected eMac USB 2.0 units tend to fall in these ranges of serial numbers:
- G8412xxxxxx- G8520xxxxxx
- YM412xxxxxx - YM520xxxxxx
- VM440xxxxxx - VM516xxxxxx
Your symptoms sound like it could be this model. Apple had a repair extension program to replace defective logic boards but it expired over two years ago. Today, the fix is not economically feasible.
Guess I will have to go and look at PCs.
Consider the 13-inch MacBook Pro I linked. I had a 15-incher that I bough about six years ago because I did not think anything smaller would work for my old eyeballs. However, in June I bought the 13-inch as a refurb directly from Apple for only about US$1000 and, as long as I remember to put on my computer glasses, I have no trouble with the smaller screen.
Based of the lack of issues with that model reported in the MacBook Pro forums, it seems a reliable model. My son bought one for university as a refurb. I liked it a lot so bought mine in June. A friend who had been using 1998-vintage Powerbooks until this year liked mine so much that he bought two. You can use them like a desktop computer. When I'm not using mine away for home, I have a wireless keyboard and mouse set up for it so it's very much like a desktop until until I hit the road.