How to set status codes such as 401 and 403?

Setting a 404 in WordPress can be done quite easily using the set_404() method of WP_Query and the status_header() function. Here’s an example taken from WP::handle_404():

$wp_query->set_404();
status_header( 404 );
nocache_headers();

If I wanted my tag archive to be accessible to authenticated users only, setting a 404 page for non-authenticated users wouldn’t make sense because the resource does exist, it’s just not available to logged-out users.

My question

How would I set a 401 in WordPress?

The same way you would in any other PHP program:

header("HTTP/1.1 401 Unauthorized");
exit;

WordPress has special handling for 404 because there’s a 404 template. Usually if something happens such as a 401, there is no special template, but you can make use of wp_die which will show a message in a simple UI

header("HTTP/1.1 401 Unauthorized");
wp_die( 'forbidden' );

This will show the message in a white rectangle with rounded edges and sans-serif text on a gray background, then call exit/die