To me, it’s looking very like a memory problem, as this is usually the symptom of bad memory. As you mention you’ve tried the RAM sticks individually, it’s looking like both modules are at fault. Could you try running Microsoft’s Memtest (link here)? After an hour of running, it should still say “No errors” at the bottom. If it returns hex digits, e.g. one or more expected and actual results, then the RAM is at fault.
If the memory is fine, try restoring the BIOS settings to their default values, in case some overclocking / memory setting has been tweaked at some stage. When you go in the BIOS setup, there is usually a key mentioned which to press to load default settings.
Next, try disconnecting everything internal and external except for the vital components, i.e. keyboard, mouse, display, graphics card and hard disk. Then check if the problem occurs. I’ve once had a faulty USB ISDN adapter that caused my PC to randomly BSOD and reboot.
Finally, try another IDE cable between the hard disk and motherboard.