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);
}