This is extremely odd behavior. I would apply all updates to this problem computer (XP service pack 3) and update the network interface card (NIC) drivers. You will probably have to download these updates to another computer and use a CD/DVD or USB drive to copy the files to the problem computer.
You should also scan the computer for malware. The Microsoft Malicious Software Removal Tool is a good tool to start with.
Did this computer ever work on a LAN? Do you remember when this problem started? If so, then maybe do a system restore to a date prior to the problem starting. If the problem continues after updating the system, then you are probably at the point of reinstalling the system from original media.