I administer a Mac OS X Server email server and it is down. Can you help? I hope this is a stupid question [with an obvious answer that I am missing] ...

My email services have suddenly stopped working and nothing I do seems to affect the problem.

Existing mail accounts are broken.  New mail accounts can't be created because the account "fails to access the [specified email server].  This happens on my local servers and on external servers.

This happens on both of my local machines, one running Leopard Server and the other running Snow Leopard Server.

The Network Utility ping function succeeds with "support.apple.com" but fails on "apple.com"

How could one work, but not the other?  Is this even related to my problem?
The failed pings are completely unrelated to your mail issue. It is true that apple.com is blocking icmp traffic (ping uses icmp) and the endpoint at support.apple.com is not blocking this traffic. This is possible because each of those hostnames point to different servers/endpoints with different filtering rules on each.