Do they just keep trying to produce colder and colder temperatures, and keep coming against this barrier.
Gasses follow certain laws, which physicists have deduced.
Double the pressure, and the volume of gas becomes half. This reduction of volume happens in a very predictable way. And you can deduce how cold you would have to make the gas, in order for the volume to be zero. This is the same temperature for all quantities of all gasses. A gas cannot have a zero volume.