EMac shuts down while on-line

Question

I'm visiting relatives in Ukraine, and they have an odd problem with their eMac that I've been trying to fix. The eMac will, suddenly, with no warning, shut down while on-line. First, the windows and toolbars all disappear, leaving only the desktop photo. Then, seconds later, that disappears, too, and the computer shuts off.
This seems to happen only while using Gmail or Yahoo mail; earthlink web mail hasn't triggered it. (And I haven't tried any other web mail sites.)
Any suggestions?
SO far I have:
repaired disk permissions
verified the disk
run the Apple hardware test
upgraded to 10.4.10
installed iLife '08
installed and tried Firefox
Thanks.

Answer 1

Good idea, trying the test user! That the test user ran OK helps narrow done the troubleshooting steps, although it still leaves a lot of open questions.
The flashing question mark followed by eventual starting indicates that the startup disk PRAM setting is lost or corrupted. Open System Preferences -> Startup Disk and reset the startup volume to be the hard drive's OS X volume.
The missing hard drive in Disk Utility sounds bad, but if the hard drive were corrupted, it shouldn't be starting up and your test user account shouldn't be running. If you're braver than I am, you can try remotely talking your family through using the Unix file system check utility as described in Using Disk Utility and fsck.
That the test user you created had no problems would seem point to either a conflicting third-party add-on, or a damaged hard drive that's being triggered when Spotlight runs in the background. The latter is unlikely since your test account ran OK. Have your family try Starting Up in Safe Mode. If the Mac is stable in Safe Mode, that leaves the process of determining what third-party add-on is causing the problem (talking someone through that sort of process remotely gives one a whole different perspective on the life of call center tech support staff).
The basic idea is to move items out of the /Library/StartupItems and /Library/PreferencePanes folders and into a holding folder, retarting the computer, and seeing if the Mac is stable or not. After a particular item is identified as a likely culprit, it can either be deleted or, if the particualr function is really needed, the publisher's website checked for updates.
Good luck!