How can I make Nginx return HTTP 503 when my proxied app server is down?

You can do so by setting an error page.

error_page 502 =503 /maintenance.html

With just the contents of Martin Fjordvald's answer I was still getting 502, I didn't have a real maintenance.html page, but this seems to work:

    location / {
        error_page 502 =503 /maintenance.html;
        include proxy_params;
        proxy_pass http://unix:/var/run/my.sock;
    }
    location /maintenance.html {
        return 503;
    }