You should realize that once you choose to re-initialize the drive, the only way you're going to get your files back is through "data recovery".
There are several data recovery apps out there.
One of the best is called "Data Rescue 3".
Your problem is a common one amongst folks who trust their data to ONLY ONE drive.
One drive IS NOT ENOUGH.
You need AT LEAST TWO COPIES of anything that's important to you.
If you keep things on only one drive, and it starts having problems -- well, you know about that now.
Suggestions:
- buy a USB3/SATA docking station (about $25. Go to amazon and enter "usb sata dock" in the search box, and you will see many choices. I use and recommend plugable.com as a satisfied, paying customer).
- buy a "bare drive" (2.5" laptop drive is fine)
- put the drive into the dock, connect it, use Disk Utility to initialize it.
You now have a "second storage source". You can use this to recover files from the old (non-functioning) external drive, unless there is a hardware problem that keeps the drive from working.
More thoughts:
The problem with your external drive may be hardware, or it may be software (corrupted directory).
A hardware problem is usually beyond the realm of the end-user. In this case, you really need a "data recovery firm" to open the drive and access the platters, but be aware that this is -VERY EXPENSIVE-, and can run into the thousands of dollars. Is the data worth that much to you?
If it's a software problem, you might try DiskWarrior on the drive first. DW may work, or it may not.
If DW -DOES NOT- work, there could be an alternative method to get some or all of the data back, but it's not for the feint-hearted. You might consider re-initializing the drive (it will now appear to be "empty", but with a "clean, working" directory). Then, use a data recovery app like DataRescue3 to access and "scavenge" the drive. DR3 will ignore the directory, and go "right to the platters" to scrounge around. It can find pieces of files, collect all the pieces, and then reconstruct them into usable files on the "scratch disk" (that's why you need another drive). You will lose folder hierarchies and file names, but you can get the actual _data_ back.
The above scenario worked for me when nothing else would mount a bad partition I had (directory corruption). It's a last-ditch effort, but it _can_ work.
I'd try DiskWarrior first. But if that doesn't help, you're going to need to go the DataRescue route, or face the fact that the drive itself has hardware problems and that NO software recovery is possible....