If it hasn't been plugged into a power mains outlet for some time, your eMac's PRAM battery is dead and won't let your eMac start.
You'll need to replace it.
It is located underneath the eMac protected by a secured cover.
It is the same location as the eMac's RAM sticks
You need to unplug the eMac, first.
Next turn over the eMac and make sure you protect the eMac's screen with a soft towel or blanket.
Unscrew the access cover.
You'll have to remove the RAM to get at the battery. Replace with new battery, reinstall the RAM sticks, plug the eMac back in and push the power button and see if this fixed the issue.
Cheapest place to get one is here.
http://eshop.macsales.com/item/Newer%20Technology/BAA36VPRAM/
If this is all that is wrong with your eMac, then it should start up as normal.
Message was edited by: MichelPM