Well, let's see...
It sounds like something is "timing out." So go to Start Settings Control Panel Power Options and set all the options to "never." This means that no devices will ever turn off. I set mine to 5 minutes or so to save the monitor and save power. Try that one.
You could try running SE from the CD. Don't install Windows to the hard drive, run it from the CD. See if that cures it. If that doesn't do it, try this:
Get a DOS prompt, either from Start, Accessories, or boot from a Win98 start-up floppy disk. Now, do "fdisk"; delete all the partitions; and then format the hard drive. After this, do "fdisk /mbr" to delete the master boot record. Your hard drive should now be completely blank. You have to ask yourself, "Does my machine auto boot from a CD?" ( probably does, but check... knowing this saves time )
Now, turn on the power switch; when the company logo comes up, hit "F8" ( top of keyboard ) to get to your ROM BIOS. Notice the date of manufacture; is it old ? Like 1999 or before ? If so, go the BIOS manufacturer's website and obtain an upgrade and save it to the desktop. This resets an erasable PROM on the BIOS chip. I've done this without any noticeable harm, but be careful; you are altering basic hardware configuration !!
Good luck !!!!