Can a unmounted volume be repaired?

You might want to try and use the disk utility feature by booting up using the option to choose a different drive before the operating system loads. I have had success working with drives that seem to have issues with the OS active and running. The other thing you might want to check is instead of just viewing the volume to choose to view the devices as well. If the device is available and the volume won’t mount you can try first aid on the device and see if that fixes the issue to allow the volume to mount. Every bug can be different depending on the issue but I hope this helps.

You cannot run Disk Utility to repair the boot volume while you are booted on that volume.

You can boot into Recovery mode, and run Disk Utility on the boot volume from there.

You should also be able to run Disk Utility against external disks from Recover mode as well, and since Recovery mode will not mount them automatically, you should not have any issues with them being mounted.

Does Disk Utility show it as unmounted?

Have you tried to unmount the volume and run Disk Utility more than one time with the same result?