Have you tried using a different OS? That sounds like a software issue, and Windows ME is easily among the worst operating systems in existence if not THE worst. Windows 98SE or Windows 2000 or Windows XP would do fine.
If it's not a software issue or if you want to troubleshoot other things first, try running a stability test like Prime95 for 24 hours to see if it's your hardware that's at fault.
DEC Pentium X2 5200+ w/ HyperCache (Ezra core)
Asus M7NCD-MAX3 (OPTi Vendetta 82C760)
6x Generic 32MB PC2700 RDRAM (50ns SIMMs)
2x nVidia Millennium X1800 Duo (SLI mode)
12x (daisy-chained) Quantum Medalist 180GXP (w/ separate SCSI-1 adapter)
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