It's a joke, a sarcastic joke. Hospitals have emergency rooms. A "predicament" is simply an awkward situation. The receptionist is talking about something unimportant. Chandler wants her to pay attention. Chandler says "It's kind of an emergency." He then realizes that it is an emergency room, everyone who comes there is having an emergency... and therefore the receptionist should always give arriving patients first priority. He means three things. 1) This isn't some unimportant little predicament, 2) this is an emergency, and 3) I SHOULDN'T NEED TO TELL YOU THAT." He says this in a sarcastic way. There is no such thing as a "predicament room." He pretends there is. If this were just an unimportant predicament "we'd be in the predicament room." It is his way of saying, indirectly, "you should know that when anyone arrives here it is an emergency and you should pay attention to them."

Chandler is being sarcastic. He thinks it should be obvious to the nurse that they have an emergency because they came to the emergency room instead of the (nonexistent) "predicament room", which would theoretically be for less serious problems.