ATC systems basically have 2 of everything, from radio systems and radar systems to flight data processing systems, flow managment systems, data communication systems and power supply systems, with centres having 2 connections to the national grid and backup battery supplies. What is usually of great concern is common point of failure, the one point in the system that will render it useless. this sometimes gets very difficult as a in older equipment rooms, air conditioning or the odd water pipe that can spray both systems and have a disasterous effect. Hence newer centres have 2 equipment rooms with fire walls etc between them.
Computer systems generally have 2 servers, disk sets, data links etc, the number of clients on a system can be rationalised or switched between either system, on a bad day only half the clients should be lost and ATC may have some problems but at least they have some functionallity,
The only real common point is sometimes the software, if the same bad data is passed to either servers and causes software problems, which has been known to happen, then we really have a problem, there is currently no way we can develop independant software for duplicated hardware. but we do have systems for trapping bad data, or at least 2.
More on the safety critical side is actually knowing the equipment is doing what its supposed to be doing. Think about an ILS being in error , its actully better off being switched off. Care is taken to ensure what is recieved is what was actually sent, and not slightly different.
Good luck
Rick