I’ve searched the forums and haven’t found this question asked anywhere else.
Is there any way to set the site to go offline at a specific time the same way we can with publishing and unpublishing resources?
Not easily. You might be able to use a cron job to toggle the site_status field in the modx_system_settings table in the database (0/1).
I guess you could also have a snippet call in your template(s) that did that or a plugin that listened for OnWebPagePreRender and redirected to a "Site Offline" page if the time wasn’t right.