It is possible to set a browser window to fullscreen using the HTML5 Fullscreen API. I'm not sure if the website you mentioned is using this approach though.
This is not a standard yet and thus will need vendor prefixes. You can check the level of browser support at CANIUSE and HTML5 Rocks has a short tutorial Fullscreen API.
Edit: Updated answer with code.
As I suspected, the website uses the HTML5 Fullscreen API. Refer to the code below on the page which_popup.php
function popup_(url, isOnClick) {
var is_popup_activated_domain = getCookieA('haaretz');
if (is_popup_activated_domain == "" && the_cookie == "") {
if (isOnClick) {
setCookieA('haaretz', 'yes', 14400);
the_cookie = 'yes';
}
var chrome_full = false;
if (document.documentElement.webkitRequestFullscreen) {
document.documentElement.webkitRequestFullscreen(Element.ALLOW_KEYBOARD_INPUT);
chrome_full = true;
}
//....
if (chrome_full) {
document.webkitCancelFullScreen();
}
}
}
It basically listens to a click event on the text, sets the browser to fullscreen and then opens the popup, so the popup also enters fullscreen. The api methods that you need are webkitRequestFullscreen to enable fullscreen mode and webkitCancelFullScreen to exit the fullscreen mode.