What do you do when you start to install macOS Monterey and your computer totally locks up?

Shut off the Mac by depressing the Power button until it shuts down.

Then boot to Safe mode. That boot takes minutes

How to use safe mode on your Mac - Apple Support

If still problem reinstall boot to Recovery and repair/First Aid the startup disk. If still problems reinstall macOS.

How to reinstall macOS - Apple Support

As mentioned by previous posting first.

Should the issue be lack of empty space as per Apple Requirements   >> If upgrading from macOS Sierra or later, your Mac needs 26GB of available storage to upgrade. If upgrading from an earlier release, your Mac needs up to 44GB of available storage.  and the drive is completely full, one may have to explore Transfer files between two Mac computers using target disk mode