How to repair or back up external hard drive when disk utility can’t repair it

If you do not care about loosing the data already on it, try changing the partition scheme instead of formatting it.

Select the drive like in your screen shot, then click on Partition tab on the right, then select 1 partition from the pop-up menu on top of the graph. Then click Options at the bottom and select GUID (the first option) in the next window, close it, and then click Apply in the main window. You can also set the format of the partition before you click apply. It should already have selected Mac OS Extended. All this assumes of course that you wish to use this drive with a Mac and only a Mac… if you want to use it on a PC and a Mac, you'll have to select other options, but still possible.

If that still doesn't work, it's probable the drive is physically damaged.

What you have right now is a Windows Partition scheme and a FAT formatted partition. I could be mistaken, but I think FAT32 does not work on OS X. I'm only 50% sure though...

There is a chance to repair it with command line (in Terminal) with the terminal utility as described in Apple's support docs, shown below:

Use the command line and the fsck_hfs -l command.

Start up your computer and log in as an administrator.

Open Terminal (/Applications/Utilities).

At the prompt, type the following command and then press Return to determine your filesytem ID:

df -hl

Look for some lines of text that look like this:

Filesystem Size Used Avail Capacity Mounted on
/dev/disk0s3 37G 20G 17G 55% /
/dev/disk0s5 37G 37G 641M 98% /Volumes/Storage

Make a note of the External Disk "disk" name that appears after /dev/, such as "/dev/disk0s3." This is your filesystem ID for your External volume.

At the prompt, type the following command and then press Return:

df -hl

Then type the following command, where "disk1" is your filesystem ID you noted in step 4, then press Return:

sudo fsck_hfs -l /dev/disk1

When prompted, enter your admin password, then press Return to begin the verification.

You should see messages like these during the disk check:

** /dev/rdisk0s3 (NO WRITE)
** Root file system
** Checking HFS Plus volume.
** Checking Extents Overflow file.
** Checking Catalog file.
** Checking multi-linked files.
** Checking Catalog hierarchy.
** Checking Extended Attributes file.
** Checking volume bitmap.
** Checking volume information.
** The volume Macintosh HD appears to be OK.