First, as there is a large group of eMacs that either have or will develop fatal logic board issues, we need to see if yours in the that group. Based on your symptoms, I'm guessing it is among the dead or soon-to-die.
Use this Apple article:
to find your serial number. Please DO NOT post you number here.
Now compare your serial number to these lists of ranges of affected eMacs:
- G8412xxxxxx- G8520xxxxxx
- YM412xxxxxx - YM520xxxxxx
- VM440xxxxxx - VM516xxxxxx
Let us know what you find. If it is among the affected, fixing the issue today is not economical feasible†. The good news is that the hard drive is seldom affected, meaning you can remove the HD and attach it to another Mac to recover files.
Note: If your computer is NOT in those ranges, please use the article in the first link to tell us more about your eMac; there were four sub-models---the table in that article will tell you which you have. Please share what it says about your model. That will help is help you.
†---unless you live within driving distanceo of Eugene, Oregon USA