No matter what link you take to leave, upon returning to home page, ALL links take you nowhere!

Bit of a convoluted complex set-up for a pop up but try adding the else {} block below shown in bold to your jquery script. What it is saying is if the specified time HAS NOT elapsed when revisiting the website then remove the #mask and #boxes <divs>, which has already been pointed out are only hidden from view, but are still sitting ontop of your content, denying access to any user involvement. Usually you would always set those in the css as display: none; by default but in your case given the messy implemention it doesnt work, a lack of understanding by whoever produced the original code.

if (now - lastTime > 4*60*60*1000) {
//4 hours 4*60*60*1000
//Set heigth and width to mask to fill up the whole screen
$('#mask').css({'width':maskWidth,'height':maskHeight});

//transition effect
$('#mask').fadeIn(500);
$('#mask').fadeTo("slow",0.9);

//Get the window height and width
var winH = $(window).height();
var winW = $(window).width();

//Set the popup window to center
$(id).css('top', winH/2-$(id).height()/2);
$(id).css('left', winW/2-$(id).width()/2);

//transition effect
$(id).fadeIn(2000);
}
else {
$('#mask').remove();
$('#boxes').remove();
}
localStorage['lastTime'] = ""+now;

Sounds to me like you may have two separate home pages in your site.

Verify that the links on your internal pages (the ones that are returning you to the home page) have the exact file path and name of the home page, right down to the case structure.

INDEX.html and index.html are the same page to your local OS, but on a server, they're two different files so a link to one won't go to the other. Also look at the file extension. Starting at index.php, clicking to a secondary page, then clicking a link to index.html can also cause the issue you're seeing.

A link to your site would take out the guesswork though.