If you can ping other sites (like freerepublic.com) by number, but can't ping your DNS servers by number, then they went down. Contact your ISP or whomever runs them. All but the most incompetent ISP will be well aware of any such DNS server crash. You need at least one DNS server to translate the human readable URLS (such as freerepublic.com) into numeric IP addresses (such as 209.157.64.200).
If you can ping at least one of your two DNS servers (most folks have two) by numeric IP address, but not ping any site by name, then the hardware system that runs the DNS server software is up, but for some reason (difficult to debug from here) you aren't able to use that service, perhaps because that DNS server software isn't running on that hardware system like it should be.
I do believe that the problem lies with the W2K machine's ability to contact the DNS servers. All the other clients here, they are all on Win9x, are operating without any loss of service. When the W2K machine's internet access goes down, you receive nothing but a DNS error page when using URLS, but by typing FR's IP address in the address bar, the page comes up.
I'm away from the machines until Monday. I will spend what time I can researching my ISP's DNS server addresses and W2K's handling of DNS. Unfortunately, the problem is intermittent, occurs at random times, and resolves itself after a period of anywhere from 1 to 4 hours.
To add even more difficulty to the situation, the machines are in almost constant use during business hours limiting the time I can spend trying to diagnose the situation. The fact that I have only an elementary understanding of networking terminology and protocols adds to the level of frustration. </whine
Thanks again for responding.