My first thought is that you've got a faulty ethernet card, but try this first. When your cable is unplugged, open Konsole and use
(as user, not root). You'll get a lot of information, what you're looking for is the two lines near the top that say Mem and Swap. There will be a segment on each of those lines that says "[somenumber]k free." This is saying how much memory and swap space you have free.
Keep an eye on that and plug in the network cable, and see if the number drops to zero when it starts hanging.
And right above Mem is a line that starts with CPU:, followed by a bunch of percentage numbers. Do any of those hit 95-100%?
Also
While your cable is plugged in but before the computer hangs, are you able to browse the web like normal?
You said that without it being plugged in, you can work for several hours without it hanging. Does this mean that it does hang after a while anyway, but starts earlier with the network cable plugged in?