Startup disk repair

Good evening,

To repair the startup disk, you just need to start up while holding down the cmd and R keys to boot into the recovery partition. Release the keys when you reach the utilities window, launch Disk Utility/select the hard drive (HD) and click on SOS. Depending on the message, run SOS again; you should get a message indicating that your disk is okay, but beware, if your hard drive is too full (less than 10% of its capacity free), you will receive an error message.
For your formatting, here’s how to do it from Sierra and later.

From Sierra?


Launch “Applications / Utilities / Disk Utility”
Select the hard drive to format/wipe.
Click on the “Erase” button. 

In the window that appears: 
Give a name to the hard drive. 
For the other two fields (format and scheme), there are three possible cases:

Case 1: For a disk used only on Mac: 
- “Format” = “Mac OS Extended (Journaled)” 
- “Scheme” = “GUID Partition Map”

If you plan to use these external disks on both Mac and Windows PC

Case 3: For a disk used on both PC and Mac 
- “Format” = “exFAT” 
- “Scheme” = “Master Boot Record (MBR)” 

Finally, click on “Erase” to start formatting the disk. 

Once the disk is wiped, you can create partitions by clicking on the “Partition” button. 

© Ritchi

--
When it's urgent, it’s already too late (Talleyrand)
When all else fails, read the manual ;)

Thank you very much for this clear and precise response. I couldn't find an equivalent when searching online before asking my question here.

For my external drives: when I want to erase the volume, I do not have access to "GUID Partition Table," only to the format. This lack is indeed indicated to me when I want to install Mojave on the external drive.

For the second solution, since my hard drive is only at 50% capacity, I will create a partition on which Mojave will be installed, and then I will boot from this newly created volume, which allows me to repair the startup disk of the initial volume. Is that correct? How much space is needed on the internal disk just for Mojave?

There is a good chance that your disk is dead or heavily damaged, too much to be repaired by the disk utility, not powerful enough for major failures! You would need to use a utility like DiskWarrior, booting from its disk, but this utility is not cheap.
I think you should consider replacement.

--
When it’s urgent, it’s already too late (Talleyrand)
When all else fails, read the manual ;)

Thank you for getting back to me, even though I would have obviously preferred a different answer.
No repairman on site, so a replacement at Apple means 400 miles for me, if I go for 2 trips (1 to drop off, 1 to pick up).
No change in the problem, so I'm just dragging it out...
It's true that Disk Warrior is quite expensive, but I would almost consider trying it, is it easy to use? If it’s only in English, I’m a bit hesitant given my level when it comes to topics other than just the weather.