It should be possible, assumming the two computers have exactly the same version of Windows. It really depends on whether you're confident using DOS I think.
On the computer that's ok, build a DOS system (i.e. bootable) floppy disk (or CD if either computer hasn't got a floppy drive). Use that disk to boot up the computer with the problem and copy the boot files off the boot disk using DOS. Some of the files will be hidden.
Use a CD-RW to avoid wasting CD-R's with attempts that don't work.
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Another idea, boot up the computer with the problem with a Windows installation CD. It might be able to repair things - BUT there is a risk that it might also wipe everything, so probably not advisable unless you've got a recent backup of anything you care about.
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You might need to fiddle with the BIOS to make it boot from floppy or CD instead of the hard disk.
HTH, John