You have the oldest generation of eMac. I got rid of the one like that years ago, but still have 2 USB-2 eMacs. Still in common use. (They both get a little "grumpy" when I wake them up after several days (sometimes months) of sleep.
The failures that I have had over the years were mostly the well-known swelling/leaking capacitors. Be sure to open your eMac, to give you access (and a close look) at the Down Converter board. That's where the issues happen the most. If you see any problems with the capactors, pretty simple to fix, by replacing the capacitors. It's been a while since I last did that (maybe 15 years) and no longer have the parts for the oldest eMacs (such as yours) But, more common, at least for me, is broken power buttons (just stop clicking when pressed). A quick ohmmeter check will help you decide if the button has died. I used to have a couple of spares, but, again, I don't have anything now for the oldest eMacs (the two I have are the newer generation, and don't share any parts with your model eMac.)--. I did find the original part number for your power button, if you need it... Apple part number 922-5075. Might help, might not. Worth the try, I think.