A few suggestions
A) What operating system are you running, exactly? Windows XP/Vista/7 should be able to recover most forms of logical drive failures. If you go into "My Computer" and right click on the drive, go to "Properties" then "Tools". Then select "Check Now". This should pull a "Check Disk XXX" where XXX is the name/drive letter of the drive. Select "Automatically Fix Files" as well "Scan for and attempt recovery of bad sectors". This will fix most logical and some basic hardware failures. With any luck you should be able to correct most of the drive and get most of the files off. You may lose some files that are beyond recovery, but it shouldn't lock up windows as the drive is unmounted while it's doing that.
B) Another possibility would be to do the copy via command prompt. This is a bit more...persnickety shall we say? But it works most of the time for me. The following commands assume that the external drive is F: and you are wanting to copy the files to an directory named Backup on the C drive.
xcopy f: c:\backup /S /V /C /I
For those wondering the arguments translate to:
/S = copy all subdirectories except for empty ones
/V = verify each file after copying
/C = continue even if errors occur (this stops it from dropping the whole copy process on an error)
/I = if copying multiple files and the destination does not exist, assume destination must be a directory and create it
I would encourage running this after A as doing so should prevent system lockups at the state.
C) Hopefully, one of the above methods works as they are free. Options beyond that generally cost money to one degree or another. The cheaper methods are less certain ranging up to more expensive for more or less guaranteed recovery. There are a number of cheap utilities available on the net, alas, I haven't used any of them in some time, so I can't speak to their effectiveness personally. Doing a search for "Data Recovery Software" should find some utilities that could help, and most run in the sub $100 range generally. The more expensive ones are generally intended for computer shops/offices that support numerous systems and require professional line software. Alternatively, call some of the local shops in your area and see if any of them do data recovery services. The last option in this chain would be high end, professional data recovery. Typically these services start in the low $200 range but can quickly climb into the thousands.
In the end, it's really a matter of "what is your data worth to you?" Hopefully, A or B works for you and you're able to recover most, if not all of your data for free. If you're still having problems, let us know details and we may be able to come up with more options.
What operating system (Windows XP, Vista, Windows 7)?
What kind of PC (Local shop/home build, Dell Latitude 620, HP Pavilion PC)?
What make/model of external drive and what size?