Agreeing again with Lumpy, and the error page I see when hitting your site with an invalid path like https://forum.homestuck.xyz/404
shows the web server's 404 page, instead of phpBB's "The requested page could not be found" message. So you can just set aside that previous in-depth reply for now, because it's not what's happening on your site yet...
Since your HTTP responses suggest an Apache web server, the .htaccess file at the root of the phpBB installation (the same directory where phpBB's config.php exists) is what should have caused the attempt to access this invalid path to be routed through phpBB's app.php. Like if we do the same thing here on https://www.phpbb.com/community/404
So for the situation you're literally in right now, using your hosting provider's instructions for how they support adding custom 404 error pages (or simply creating those error page references in the .htaccess file yourself) is how you could make a more customized error page. i.e. You're not getting any of that phpBB-specific behavior we were expecting.
But in order to make phpBB work the way it was intended, and/or the way you might require it to work with some future extensions, you should probably investigate and fix why phpBB's recommended and included .htaccess rules are not in effect already. At which point you will be in the position of "now do I want to customize The requested page could not be found
page that phpBB shows."
If you're not sure why phpBB's .htaccess isn't already in effect, perhaps just post the contents of your current .htaccess file (the one in the same directory as phpBB's config.php) between two
tags here. It may be that some additional rules added to that file have inadvertently caused the phpBB default rules to become skipped, or similar.