Radio and radar failures at an ATC unit are fairly uncommon, although I did experience several of each! I don't know what equipment is available now, but standby R/T was always provided where I worked. However, if the radar goes off you can get busy.
Comms failure on an aircraft presents few problems to a radar unit - just a matter of sitting and watching it and provide separation.
Emergencies - or what are referred to as such - happen more often that you might think. The most common are medical emergencies when a passenger or crew member requires urgent medical assistance and aircraft occasionally experience technical problems which require priority landings, but none present problems for a trained controller.