Twig: How to show static text in the template to the Anonymous User?
Add a page--403.html.twig template to your theme.
See this change record: https://www.drupal.org/node/2960810
If you only want to change the default message you don't need a template. You can configure a node as access denied message in /admin/config/system/site-information.
Caveat for the non-template solution (as commented by @FreeRadical): If you restrict access to all nodes you have to find a different type of content. Could be a small module providing a custom controller which returns the static text.
BTW the themed page--403.html.twig is built in a subrequest which only job is to return a themed page with blocks. It's too late to make decisions on what to do with a 403 exception at this point. If you want to replace the core handling of a 403 exception (returning a themed subrequest) use an exception subscriber with default priority. See https://drupal.stackexchange.com/a/245211/47547