Possible causes (in descending order, with what I consider the most likely cause first, followed by the second most likely, etc.):
- VRAM (video memory) failure; the logic board is toast. You'll either have to replace it yourself (not necessarily easy if you've never done it before, and the eMac isn't the safest machine in the world to work on, given the voltages the CRT can retain, even after its powered off), have it replaced for you, or scrap the machine.
- GPU (graphics processor) failure; the logic board is toast. See #1 for details
- DAC (digital-analog converter) failure; depending on how the eMac is set up, either the logic board, the crt's board, or both are toast. See #1.
- Something (most likely the GPU) is overheating. Either due to dust buildup (in which case blowing the system out with canned air or an air compressor may fix it) or a fan failure.
- RAM (the system RAM) is toast. This you can replace yourself easily.
- CPU is toast. See #1.
- Some software issue that I'm not aware of that may be cleared up by a PRAM reset or something else.
My bet is on #1, 2, or 3 to be honest. It's very possible that it's #4, however, IF one condition is met: IF it only does this after it has been running for a while, #4 is most likely the source of your problem. If that's the case, that overheating could have lead to permanent damage of the GPU, VRAM, CPU, or any number of other components, which could continue to cause this after the overheating problem was solved. But if it does this from a cold boot-say, after it's been sitting turned off overnight, then #4 is basically ruled out entirely.
I'd recommend blowing it out at least, since it costs next-to-nothing and might be worth a shot. If that doesn't work, and nobody has any software suggestions... I'd recommend you consider whether this is a project you want to try to diagnose and repair yourself (probably not something I'd recommend on an eMac if you've never done it before), or alternatively if it's worth the cost to have it diagnosed and repaired (which, if it needs a logic board, could end up being more than the machine is worth unless you source a used board on ebay or whatever and have that installed), or if it's just time to recycle it and buy a new machine.