How do I fix my 2013 Mac stuck in restart loop?

The ordinary advice would be to launch Disk Utility inside MacOS and use its FirstAid/RepairDisk function to try to fix up the Directory of boot drive. but if the Mac will not boot normally, you are forced to do that from Recovery.

I think you said your Mac would not start up normally, is that right?

But your boot drive is so confused it can not find the portion to the drive where the Accounts are stored, so it can not find an Admin account to use to authorize Recovery to run from the boot drive.

Next step is this:

If you can’t start up from macOS Recovery (Intel-based Mac)

To help ensure that your Mac can recognize the keys you’re pressing at startup:

• If you’re using a Mac laptop, use its built-in keyboard instead of an external keyboard.
• If you’re using an external keyboard, you might need to wait a few seconds before pressing any keys to give your Mac time to recognize the keyboard as it starts up. Some keyboards have a status light that flashes briefly when the keyboard is ready to use.
• If you’re using your keyboard wirelessly, plug it in to your Mac, if possible. Or try a wired keyboard.
• If your keyboard is made for PC, such as one that has a Windows logo, try a keyboard made for Mac.

If your Mac tries to start up from its built-in Recovery system but can’t, it should automatically show a spinning globe as it tries to start up from Recovery over the internet. Or you can force Internet Recovery by pressing either of the following combinations at startup, instead of Command-R. (Each combination also affects the version of macOS offered when reinstalling macOS from Recovery.)

• Command-Option-R
• Shift-Command-Option-R

If the spinning globe includes a warning symbol (exclamation point), startup from Internet Recovery was unsuccessful:

FROM: How to start up from macOS Recovery - Apple Support

NB>> Please let Readers know if you care able (or not) to make progress by using one of those other command-key sequences.

is this a Mac Pro 2013 dark cylinder?

or a MacBook Pro 2013 with 13-in display?

Do you remember what MacOS you were running?

Do you have another Mac available? what model-year?