How do I turn a 404 page into an automatic search with the info from the url?

This can be done in two easy steps.

1. Upgrade your site search with a plug in, such as Relevanssi (optional).

2. Add the following code in functions.php or via Code Snippets (thx to @Howdy_McGee and Russell Jamieson for ideas!)

function wpse_204310() {

    global $wp_query;

    if( is_404()
        && !is_robots() 
        && !is_feed() 
        && !is_trackback()
      ) {
        $uri    = $_SERVER['REQUEST_URI'];
        $clean  = str_replace( "https://wordpress.stackexchange.com/", "%20", $uri );
        $clean2 = str_replace( "-", "%20", $clean ); 
        wp_redirect( home_url( "/?s={$clean2}" ) );
        exit();
    }
}

add_action( 'template_redirect', 'wpse_204310' );

That’s it. This code will parse the url for the blog title, and pass it along to your search page. All transparent to the user. This assumes a simple permalink structure, such as domain.com/post-title