When you get an "illegal operation" error message, click the "Details" button, and post back with the exact details word for word, up to but not including the memory registers.

What is an "Explorer windows button"?

Also have a look here:
Error Message When You Open a New Folder:
http://support.microsoft.com/?id=kb;en-us;259096

(1) You may be in for a long haul...
http://search.support.microsoft.com/kb/c.asp?fr=0&SD=GN&LN=EN-US MSKB
Search "Kernel32"; Windows 98; Title Only; Exact Phrase; produces over a
page full.

http://www.generation.net/~hleboeuf/errkrn32.htm
Many are here too.

http://aumha.org/a/k32fix.htm
Eshelman's page about Kernel32 errors.

Invalid Page Faults:
http://aumha.org/kbeipf.htm
http://support.microsoft.com/support/windows/topics/errormsg/kbipf.asp
http://support.microsoft.com/support/kb/articles/Q286/1/80.ASP

(2) Here is what mine looks like in Win98SE...

KERNEL32.DLL
Desc: Win32 Kernel core component
Loc:  C:\WINDOWS\SYSTEM
Size:  471,040 bytes
Mod: Friday, April 23, 1999 10:22:00 PM
Ver:   4.10.2222

If it comes to reinstalling it, that is old enough never to have been
Windows Updated. Therefore it should be recoverable with SFC.

(a) "START, Run, SFC"
(b) Bolt "Extract one file from installation disk"
(c) Enter "KERNEL32.DLL" in the "Specify..." box.
(d) Click "Start". It should offer to "Save file in  C:\WINDOWS\SYSTEM".
If not, make it so.
(e) In "Restore from", if not already pre-filled, browse to
1. Installation CD (likely the Win98 folder) or
2. The folder that has your cabs on the hard drive, likely
"C:\WINDOWS\options\cabs\".
(f) Click "OK"

It will offer to save the existing Kernel32.exe. May as well do so.
Likely it will ask that you reboot to complete the install.