I have some radiation-hardened, space-grade 8086 CPU's from Intel in my spares - wonder if they would survive? (Probably not).
I've seen the other post about building a radio too. Seriously, I wouldn't bother myself. Much better to buy one, from cheap to ridiculously expensive depending on your funds, and keep it in a safe. The knowledge of how to build and repair them might be invaluable if the 'problems' continued for a long time though.
Think though: Who do you think will be transmitting? If there is any basic governmental structure left, and they want to communicate to the public, normal consumer radio is what will be used. If it's more military or police that you are thinking of - they are more likely to have encrypted comms that you can't monitor. Radio amateurs are a very good bet, both national and international. For that, you want a good short wave/HF receiver or, better, transceiver. That's something you could make. A big aerial is a must too.
However, if you have suffered the sort of attack/event that has knocked everything out by EMP, you have some really serious issues to contend with. Got enough food? Water? Off-grid power? Fuel for generators and vehicles? Sanitation/decontamination/water purifying/medical supplies? ...defense when people start to panic as it all runs out, or just try to take advantage of the situation???
This is an area I've studied, and prepared for, for a long time, a lot.