You did not mention whether this problem was specific to the one user so I will assume that there is only one user for this machine.
If that is the case, there is a possibility that the profile for this user has become corrupt in some way.
Create another user profile on the eMac and see if the problem follows the profile change.
Like you, I don't see this as being a hardware issue, at least not before the changing accounts check.
If the problem does not follow the profile change, just recreate the users profile and send them on their way
Let us know what the result is
P
Okay, so I do hardware warranty repair work for a private college in California, and I ran into a situation with an eMac I am curious about.
The person who uses the eMac in question would have problems with programs not opening. Mac OS X would boot successfully, and for maybe 1-2 minutes apps would load normally, but then they simply would not load at all. The Dock icon would bounce, and then eventually stop, with no sign of the program having started. Any app that was started in that first minute would continue to operate just fine.
If you Safe Booted the machine, apps would continue to load successfully in perpetuity. This, to me, is like a giant flashing neon sign saying that it's a software issue, but the University's Mac "expert" swears up and down that this is a hardware issue. Either a bad hard drive or logic/motherboard.
The system was running 10.4.2 originally, and then I discovered that there was a known compatibility issue between the AV program used (Sophos) and 10.4.0-10.4.2, so I updated it to 10.4.11 and all the latest updates, however after checking back with the user after a couple of days, they reported that they were still experiencing problems.
I've done a little poking around with Google, and on Apple's website, and all I can find are a few difficult to corroborate reports of bad caps finding their way into some eMacs just like some Dell systems of the same era. This absolutely does not strike me as a bad cap issue, because safe booting works splendidly for as long as you want to use it, and the OS will boot consistently every time. It seems to me, like the AV program gets stuck trying to update itself, and then it ties up the rest of the system, except for the GUI that Apple has running at a very high nice level to give the illusion of responsiveness. I suspect, that if one was supremely patient, the apps would eventually load.
So, like I said earlier, I have been unable to find any kind of official documentation from Apple on this, and all I can find with Google are some forum postings about bad caps, and I really don't think this is a bad caps issue. However, I am willing to admit that there's a possibility I'm wrong on this, and so I'm wondering if anyone here can point me to a specific Apple KB article, or something similarly credible that details this issue and explains how it is a hardware failure. Because right now, it seems to me like it is some program the user installed that is causing problems, and while replacing the hard drive and/or logic board may work, it's an unnecessary repair, working only because it would require that the software all be reinstalled.

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic