Back-up / Retrieve Files from an external hard drive that doesn't appear in Devices

Question

Someone please help!
I am going to write the "history" from my disk utility repair here so that other's who copy and paste into search may find it:
(PLEASE SCROLL 13 ROWS DOWN)
Verify and Repair volume “Your Hard Drive Name”
Checking Journaled HFS Plus volume.
Checking extents overflow file.
Checking catalog file.
Keys out of order
Rebuilding catalog B-tree.
The volume   could not be repaired.
Volume repair complete.Updating boot support partitions for the volume as required.Error: Disk Utility can’t repair this disk. Back up as many of your files as possible, reformat the disk, and restore your backed-up files.
Ok. I would really love to reformat this disk, but I have stuff on here from film school that I CAN NOT lose
Question:
       How would I go about backing-up the disk if the image doesn't appear under devices? I went to the "Restore" tab/button, but I can not choose the hard drive in the first field. I could have sworn my hard drive was formatted for Mac, but I may have formatted it back to FAT32. I know the disk still functions because when it is powered on, it cycles with/without being connected to my MacPro, but after that, it will only cycle if connected (confirming connection). It does run again when I connect the firewire cable. Here is my equipment:
8-Core MacPro 16GB-Mem RAID
MacOSX 10.6.8 | 2 X 1TB Internal HDD
Aug-2008 15" MacBookPro
(That I used the Maxtor-External with mostly)
DEVICE IN QUESTION
500GB Maxtor One-Touch Plus
FireWire 400 6-pin Connection Bus (2 Ports)
Purchased in Early 2008 from Staples
I can fit the back-up on my 2nd HDD in my MacPro or in my other External:
2 TB G-Technology G-RAID External
1 x eSATA |  2 x FireWire 800 9-pin |  1 x USB 2.0
Self Powered DC-IN
I have had a nightmare searching for answers, so any options you might have would be tremendously helpful. I can't lose the data because I have everything I did during film school on there. That stuff is basically what you pay for when you go to film school, you pay to show what you can do and what equipment you have used. SAVE ME FELLOW APPLE PEOPLE!!!! We COMMAND, as well as CONTROL!
-RaphSai

Answer 1

Get Disk Warrior that will repair the problems without requiring that you reformat the drive - $99.00 on a bootable CD. Will not work on a FAT formatted drive.
You will need to use Drag and Drop to save your files by copying them from the bad drive to a good one.
To reprep the drive:
Drive Preparation
1. Open Disk Utility in your Utilities folder.
2. After DU loads select your hard drive (this is the entry with the mfgr.'s ID and size) from the left side list. Click on the Partition tab in the DU main window.
3. Under the Volume Scheme heading set the number of partitions from the drop down menu to one. Click on the Options button, set the partition scheme to GUID then click on the OK button. Set the format type to Mac OS Extended (Journaled.) Click on the Partition button and wait until the process has completed.
4. Select the volume you just created (this is the sub-entry under the drive entry) from the left side list. Click on the Erase tab in the DU main window.
5. Set the format type to Mac OS Extended (Journaled.) Click on the Security button, check the button for Zero Data and click on OK to return to the Erase window.
6. Click on the Erase button. The format process can take up to several hours depending upon the drive size.