Been a while since I've fooled with a win98 machine but here goes.
First off right click network neighborhood, go to properties. You're going to want to use 'client for microsoft networks' on both computers.
Then of course you'll need them on the same workgroup, it's in there properties somewhere. Default for win98 is 'WORKGROUP' so it should be right already, but make sure.
To share both of the computers files, in the network properties where you are there is a 'share files and printer' button or something along those lines. Click on that and check the 'share files' option. Go the 'My Computer,' and right click 'C' drive, go to sharing, and fully share the drive.
Next you need to make sure both computers are within the same 'IP range'. Now if you have a DSL modem going to a router, the router to the computers, the router should be automatically assigning the IP addresses to the computer within the same as the router, so there shouldn't be a problem with the computers seeing each other.
What you need to try at this point is pinging them to make sure the computers are able to reach each other. (sometimes they may be set up correctly to work and network neighborhood will not show the other computer on the network). Go to start, run, type winipcfg. From this window that pops up, hit the down arrow and make sure you're looking at the settings for your network card. There you'll be able to see the IP address(will most likely be something like 192.168.1.100), default gateway (will most likely be something like 192.168.1.1), and subnet mask (will most likely be 255.255.255.0) I believe. Go to your other computer and go to start, run, and type command and hit enter. In the ms-dos prompt that comes up, type ping 192.168.1.100 (or whatever the IP address of the other computer was). If you're able to ping it, wonderful. From here, you know your network is working correctly.
If you aren't able to see the computers in network neighborhood, another way to access files from another computer across the network is going to start, run, and typing '\\nameofothercomp\' and hitting enter. Obviously replace 'nameofothercomp' with the name the other computer uses on the network. You can find this name under the properties of network neighborhood again in the same place you found the workgroup.
Hope I didn't miss anything. Let us know your progress!
Scott