Retrieve WordPress’ the_content() with jQuery

As @G.M pointed out, you need to levy the awesome AJAX API. In the example below, we fire a request to the AJAX handler, along with the parameter action=gallery_popup.

$( "a" ).click(
    function () {
        $( ".gallery-pop-up-container" ).load( "/wp-admin/admin-ajax.php?action=gallery_popup" );
    }
);

In response to this request, WordPress will trigger an action of the same name, prefixed with wp_ajax_... (or wp_ajax_nopriv_... if the current user is not logged in).

So all we do now is hook onto said action and output the template part:

function wpse_137607_gallery_popup() {
    get_template_part( 'includes/single/gallery-pop-up.php' );
    exit;
}

add_action( 'wp_ajax_nopriv_gallery_popup', 'wpse_137607_gallery_popup' );
add_action( 'wp_ajax_gallery_popup',        'wpse_137607_gallery_popup' );

And you’re done!