Well isn't that interesting. Should have looked there earlier. Lets see if I can remember this correctly.
Ok lets go back into Network Properties in Control Panel and remove NOC Extranet Access Client Protocol and TCP/IP[Home] by highlighting them one at a time and clicking Remove. Put your 98 cd in the drive if prompted. Restart the pc.
We need to have your network adapter, a client and TCP/IP installed to connect to the net. Go back to Network in Control Panel. Put your 98 cd in the drive. Click Add below the box in Network Properties. In the next box highlight Client and in the next box select Client for Microsoft Networks and click ok or add. Next click Add again and this time select Protocol and in the next box select TCP/IP and click add or ok (can't remember which it is ). Restart the pc. If it wants to restart between adding the client and the protocol go ahead.
Hopefully you should be able to connect now. If not, go back to Network Properties and let me know what you see now. You should see TCP/IP-->Linksys LNE100TX[v5] Fast Ethernet Adapter. Highlight TCP/IP and click Properties and ensure that Obtain an IP address automatically is checked.
You are not using a router?
I'm curious, was this pc used in a work environment at one time?