And it has happened at O'Hare before.

In simple terms, there are two types of radar. Primary radar bounces signals off solid objects. There are also ground-based stations that interrogate aircraft transponders. The latter provides altitude information (called "Mode C"), the four digit code dialed into the transponder by the pilot ("Mode A"), and sometimes additional information such as aircraft registration data ("Mode S"). Around O'Hare all you need is Modes A and C, although most airliners send Mode S.

Anyway, there's more than one radar station near O'Hare. Occasionally one of them goes down. For example, you might lose primary coverage from the station closest to O'Hare. In that case there might be other stations still working, perhaps one near Midway. Controllers might still have gaps in primary radar coverage, but they could get by, possibly with some cuts in arrival/departure routes and/or increased aircraft separation. It depends on the nature and scope of the failure(s). The controllers do a very good job dealing with these situations.

By the way, there is some early talk about ending primary radar coverage in the U.S. With the invention of transponders in the 1960s, most aircraft have them by now, and all aircraft (except with special permission) operating near O'Hare (and other busy airports) must have them. They cost about $1,000 (and up) plus installation. But there are lots of aircraft without electrical systems, and transponders can break, so nobody really wants to pull this service just yet. Another reason is that "rogue" aircraft may not have their transponders turned on. There's a new system called ADS-B which is just getting rolled out (presently in Alaska and the northeastern U.S.) in which suitably equipped aircraft transmit their GPS coordinates continuously to controllers and to other aircraft, and ground stations can send information back to the aircraft such as live weather information and air traffic controller instructions. However, it's going to take years to get this system fully deployed nationally, and 99+% of aircraft don't have the equipment installed yet. And ADS-B probably doesn't address the needs of aircraft without electrical systems unless somebody comes up with battery operated ADS-B gear, if that's even technically feasible.

OK, more than you wanted to know.