This Google Apps Script will display the unread Gmail counter for your mailbox to anyone on the Internet with a simple URL.

The count is published as text but you may use other formats (ContentService.MimeType) like RSS, JSON or even CSV and feed this data into other system - like tweets or even another spreadsheet.

function doGet() {
  var cache = CacheService.getPublicCache();
  var counter = cache.get('gmail-counter');

  // Cache the counter to stay within the Apps Script quota
  if (!counter) {
    counter = GmailApp.getInboxUnreadCount();
    // Cache the counter value for 20 minutes
    cache.put('gmail-counter', counter, 1200);
  }

  // You can customize the output message
  counter = 'There are ' + counter + ' unread messages in my Gmail Inbox.';

  // Output the result in plain text format
  return ContentService.createTextOutput(counter).setMimeType(ContentService.MimeType.TEXT);
}