<?php

add_action('wp_ajax_foobar_action', 'foobar_action');
add_action('wp_ajax_nopriv_foobar_action', 'foobar_action');

function foobar_action() {
    check_ajax_referrer();

    wp_send_json((object) ['msg' => 'hello world']);
}

add_action('admin_print_scripts', function () {
    printf('<script type="text/javascript">window.custom_nonce = "%s";</script>', wp_create_nonce());
});

var msg = '';

// I'm using the whatwg-fetch polyfill and a polyfill for Promises.
fetch(ajaxurl, {
    method: 'POST',
    headers: {
        'Content-Type': 'application/x-www-form-urlencoded; charset=utf-8'
    },
    body: 'action=foobar_action&_wpnonce=' + window.custom_nonce
}).then(function (res) {
    msg = res.json().msg;
});