How do I hide an article element so only logged out users see it

You can solve your problem by using the built-in is_user_logged_in wordpress function :)

if (!is_user_logged_in()) {
  // content for non-authenticated users
}

You can do this by checking if the user is logged in as stated in the first answer. You simply need to do it the other way.

For example, if you have a link in the menu that should be hidden in case the user is logged in, just add a custom css;

.logged_in .menu-class-here{display:none;}

That is to hide a menu item. Now, if you want to restrict the access to a page if the user is logged in, you can simply do a redirect. Here's an example;

if ( is_page('slug') && is_user_logged_in() ) { // where slug is the name or slug of the custom page that you want to restrict from logged in users
   wp_redirect( 'http://www.example.com/desired-page/', 301 ); 
   exit;
}

If you want to allow access to the page but hide a certain section from logged in users, you can do something like;

if ( is_page('slug') && ! is_user_logged_in() ) {
  //add the code here that you want to show to non-logged-in users
}

UPDATE Add this to functions.php. You need to create a function to redirect and add it

function notallowed() {
  global $post;
if ( is_page('hire-the-freelancer') && is_user_logged_in() ) { // where slug is the name or slug of the custom page that you want to restrict from logged in users
   wp_redirect( 'http://www.example.com/desired-page/', 301 ); 
   exit;
}
}
add_action( 'template_redirect', 'notallowed' );