How to handle if Drupal site goes down, show a static fallback page instead?
Because Drupal is completely unresponsive when this happens, it's not something you can fix within Drupal.

Setting a static fail page is possible within your web server - most likely Nginx or Apache. If you are using a CDN, you can also configure it to display only cached versions of your pages if your server is unresponsive.

You'll have to consult your server documentation to get your answer.