Can I use the Backbone REST API client outside WordPress?

I haven’t tried this directly, but I’m pretty positive you can — after all, it’s a Javascript library. As long as you:

1. Load any required dependencies;
2. Provide any settings expected by the library;

You should be good. A quick code search brings up these results from the /wp-includes/script-loader.php file:

$scripts->add( 'wp-api', "/wp-includes/js/wp-api$suffix.js", array( 'jquery', 'backbone', 'underscore', 'wp-api-request' ), false, 1 );

So you have the main script (wp-api) and its dependencies. wp-api-request itself expects a settings object — here’s the relevant block from the same file:

$scripts->add( 'wp-api-request', "/wp-includes/js/api-request$suffix.js", array( 'jquery' ), false, 1 );
// `wpApiSettings` is also used by `wp-api`, which depends on this script.
did_action( 'init' ) && $scripts->localize(
    'wp-api-request',
    'wpApiSettings',
    array(
        'root'          => sanitize_url( get_rest_url() ),
        'nonce'         => wp_installing() ? '' : wp_create_nonce( 'wp_rest' ),
        'versionString' => 'wp/v2/',
    )
);

So there you have it — just remember to provide a wpApiSettings object with the required parameters.