With Nginx, it is possible to route a specific error to a named location using an error_page directive. For example:

location / {
    error_page 404 = @other;

    # other configuration...
}

location @other {
    proxy_pass https://www.digitalocean.com;
}

With this configuration, if a page is not found the request is proxied to https://www.digitalocean.com. Of course, in a real situation, that would likely be a service on a different port or instance altogether.