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First off, get yourself a copy of The Ultimate Boot CD then attach your drive as the master on your primary IDE and disconnect your other drive. See if you can boot from the HDD itself, if you can grab your files, RAR them up and stash them on an FTP, otherwise boot the UBCD and run the diagnostics program which matches the drives manufacturer and see what the condition the drive is in, depending on that you may have to get more in depth and use some data recovery tools.
-Maestr0
Along with everyone elses suggestions..
If the other PC would boot it... throw a network card in there and put it on a private network with the machine you want to back it up with. Share the drive of the machine you want to backup. (leave it on a test network with no internet access). Make sure the machine that you are backing it up to has up2date AV so you don't infect that PC too.
IIRC, Sometimes it depends on how the partition was created. I've had this happen when I tried to put a slave drive that was an extended partition and not primary partition. You are allowed up to 4 primary and 1 extended partitions. If you have 2 extended partitions... it won't be able to read it.