You will be using the operating system on the C: drive which may well be the cause of the issue. You could boot off of a floppy disk or cd or flash drive to get the copy done. Or there are free programmes out there that will continue to copy when it sees and error and copy whatever it can. Not that I can recall names now but a web search should find them.
How accurate a copy do you need ? Operating systems are best reinstalled anyway. Would it be enough to just copy over your "documents" ?