"Disk Utility can't repair the disk" is one of the messages you will receive when First Aid failed. And it would be followed up with the error messages like "macOS can't repair the disk. You can open or copy files on this disk, but you can't save changes to files on this disk. Back up the disk and reformat it as soon as you can." as you received.
The reason why First Aid failed is that your disk is badly damaged and is beyond the repair ability of First Aid. In that case, you need to back up your data as much as possible or recover the data using data recovery software. If you can't back up data from your external hard drive, you can use iBoysoft Data Recovery for Mac to give you help.
iBoysoft Data Recovery for Mac is a powerful tool for Mac users to recover deleted or lost data from unreadable/unmounting/formatted/corrupted external hard drives on macOS or Mac OS X. The file recovery tool for Mac is good at SD card recovery, USB data recovery.
- Free download and launch iBoysoft Data Recovery for Mac.
- Choose your external hard drive and click Search for Lost Data.
- Preview the search result and then click Recover.
After that, you can use the following methods to fix the "macOS can't repair the disk." issue.
- Run First Aid to repair your disk again
- Run FSCK command line in Single User Mode
- Reformat your external hard drive