WinMe is better a little more robust than 98 but it still has the same foundation: dos, so it is still a watermellon balancing on a stick
Have you tried booting into safemode and do scandisk ?
Or boot with your win98 startup diskette and do scandisk ?
And as Fabion said, do a thorough scandisk so bad sectors are repaired.
The best way to stop internet activity is to unplug your modem so you won't need Zonealarm for that