I guess that the logic board will need to be replaced.
You should verify that software is not causing this by creating a new user, and logging in as that new user. If the problem occurs then, hardware is the culprit, I'm guessing.
Also, try your Apple Hardware Test. This came with your eMac, either as a separate disk, or listed on the label of one of the disks, with simple instructions for booting to that test. You may get an error code, or simply see the problem exist even while booted to that test.
A shop should verify that you have the problem when an external display is attached, and should give you a repair quote.