Air pressure is a statistical measure; given a room with zero net variation in pressure, I can always find a volume in that room with positive pressure, down to measuring Brownian motion of atoms. Now try to design a measurement apparatus that can only sample a small volume to measure variations in pressure, and you can understand why (sound) volume can never go to zero.

Over meaningful statistical sampling areas, like an eardrum, say, pressure can be effectively constant, surely?

Volume can be a statistical property, like temperature.