ACF field check value of field on all other articles

The only way I can think of to do it would be using the save_post action hook. Since your article isn’t being saved through AJAX, and only when you actually save the post, that would be the best place, in my opinion.

I honestly haven’t written a lot of error messages, so this might be wonky and probably won’t work out of the gate, but I started with this article from SitePoint.

add_action( 'save_post', 'my_save_post_function' );
function my_save_function( $post_id ) {
    $error = false;

    $featured = get_posts(array(
        'meta_query' => array(
            array(
                'key' => 'featured_article',
                'compare' => '=',
                'value' => '1'
            )
        )
    ));

    if ( count($featured) > 3 ) {
        $error = new WP_Error('too_many_featured_articles', 'There are too many featured articles.');
        // You could use this error to list the titles and links to edit all of the other current featured articles.
    }

    if ($error) {
        // Here is where you would unset the ACF field, if you wanted.
    }

    return true;
}

This isn’t meant to get you through the whole thing, but hopefully it’s a good starting point for you.