What is a real-world metaphor for irrational numbers? In an effort to develop better number sense (and to create my own journey from fish to infinity), I have been going through Khan Academy's math material from the very beginning. So far, I have been able to develop strong intuitions and metaphors for most elementary mathematical ideas. I now see that arithmetic, division, negative numbers, fractions, decimals, factors, multiples, etc. all have clear grounding in reality; I have a much stronger grasp of the material.
So far, the only place my intuition has broken down is irrational numbers. It was surprising to me because it happened so suddenly, like the ground giving way. It was as if we saw this gaping hole in the ground, gingerly walked around it, and pretended it wasn't there. This bothers me.
Is there any straightforward metaphor for irrational numbers? Could you explain it to a child?

Imagine you can build chronometers. They all share a common mechanism such that one round takes the same unit of time. To fix ideas, say it is one minute. Now, you can draw any  number of ticks (indices?), regularly marked around the dial's edge. It could be $60$ (marking seconds), but you can choose $17$, or any integer. 
An irrational is a duration (expressed in your unit of time, here one minute) such that, no matter the number of rounds around the clock, you will never be able to design one chronometer (with regularly spaced ticks) for which the hand would stop exactly on one of the ticks, for this duration.
