I had a similar problem; with the message : "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"
I used Disk Utility and choose the option 'erase disk'; I choose quick erase; this worked good. (my hypothesis of the root cause - The usb disk had been used for Time Machine and this had partionined and created key sectors such that the DISK was useless anymore;)
Afterwards I did the disk first aid again and the disk is now perfect. I am currently using it full swing again.
- I remember I had a NAS with the same problem; I gave it away; the person who got it said ist was as simple as rewriting the root record :-((( my money flowed away...
Now you said 'reformat'; I did not find that option but what I did worked too.