A stair is a set of steps. Stair sounds like stare.To stare at someone is to look at someone. A staring contest is a contest involving two people looking at each other without blinking. It's a common activity kids engage in when they're bored. Construction workers build stairs. The construction workers are having a stairing contest, or a contest involving stair building. This phrase can sound like the construction workers are having a staring contest. This joke is a play the word stair.
Two construction workers had a stairing contest .