I agree with TheChas that paranoid is necessary in this job! If you have an open network, someone WILL find it and they WILL exploit it.
Now, all machines (PCs, switches, routers, etc) that talk IP have at least 1 open port (unless the machine is locked down completely). You need open ports for applications to communicate to the machine.
If a machine does not listen on any ports, then you can't interface with the machine and do anything.
For example, your Win98 machines will have TCP port 139 open (this is the file sharing port). Now, there are exploits that can be done against this port to a)gain illegal access to the machine, b)crash the machine c)format the machine d)just about anything else you can think of.
Your switches (if they are configured) could have TCP port 23 open for Telnet. If your router/firewall is not configured properly, it could be possible for a remote attacker to hack into the switches and re-configure them. Or bring them down.
The first thing you needto do is secure your perimeter defenses at the router. Set up the firewall on the router. Since TheChas stated that the firewall capabilities on that router are not very good, then you need a real firewall NOW. Get one, lock it down to prevent unauthorized access.
Then you can worry about internal-based attacks, because according to studies, the INTERNAL hacker is more dangerous than the external one.
Good luck.