I don't use Facebook (with or without 'Chat') but clicking on 'How come I don't have Facebook Chat?', here
http://www.facebook.com/help.php?page=713
shows that you need to be using either Internet Explorer 7, Firefox 2 or Safari 3 as your browser. (That assumes that you're using a PC, rather than a Mac).
As far as I'm aware, you can't use Internet Explorer 7 with Windows ME (which, incidentally, I'm still using - you're not alone!).
However, nearly all of the 'regulars', here in AB's Technology sections, ditched Internet Explorer long ago. Firefox is a thousand times better. (it's more flexible, more secure and, most importantly, pages load far, far faster).
The latest edition of Firefox (which has only recently been released) is Firefox 3. However, Windows ME is not supported. You need to use this link to download the UK version of Firefox 2.0.0.16:
http://download.mozilla.org/?product=firefox-2 .0.0.16&os=win&lang=en-GB
When you use Facebook with Firefox 2, 'chat' should automatically be available.
Alternatively, as that first link above states:
"If you�re using Internet Explorer 6.0, you can still use chat by popping out the chat window. Just click the Chat button near the top right of any page on Facebook to start chatting"
Chris