What is a real-world metaphor for irrational numbers?

Here is a physical metaphor:

Draw a circle, and prepare two sticks:

• One with the length of the circle's diameter (2r)
• One with the length of the circle's circumference (2πr)

You cannot cut both sticks into pieces of the same length, no matter what length you choose.

In other words, you cannot measure both sticks using the same measurement-unit.

You can try meters, feet, inches, miles, or even invent your own measurement-unit.

You will never be able to accomplish this, because the ratio between the sticks is irrational.

I am a historian of science and I am intrigued by this question, because I am a non-specialist in mathematics but for many years I taught classes dealing with the Pythagoreans. One thing you might consider in making irrational numbers "real" is to imagine why they were deemed so revolutionary and important by the ancient Greeks. In other words, instead of looking for an illustration of their practical reality, look instead to why they were so darned interesting. One answer that hasn't yet been addressed is the contention that shapes are a better representation of reality than numbers. An irrational number is the only way, in the language of numbers, to represent any "real" distance" that cannot be expressed as a relationship between two whole numbers. And yet we all know that such distances are in fact real, i.e. the hypotenuse of a right triangle. That reality, by contrast, is quite easily expressed visually through geometry, which is one reason Euclid spent so much time laying out the rules of geometric shapes. Remember that he was fundamentally a philosopher. One could argue that geometric shapes (to which Platonic philosophers believed all reality could be reduced, i.e. the "forms") are a better representation of reality than numbers are.

By bringing up metaphors, you've put your finger on the fascinating problem of reality raised by irrational numbers. The "whoa" moment regarding irrational numbers usually comes when we rethink what's behind the theorem we all had to memorize in school. Irrational has a literal meaning (no ratio possible), but also it suggests the cognitive dissonance between numbers and shapes, and helps to explain why forms were perceived as fundamental among some of the ancient Greeks.

Here a metaphor based on the irrationality of $\sqrt{2}$:

Every day two friends $A$ and $B$ enter a square field at one of its vertices, jog for a while, and leave by the same vertex. $A$ jogs along the boundary of the field (always in the same direction) whilst $B$ jogs along the diagonal joining the entry vertex and its opposite (changing the direction only when he/she reaches a vertex).

$A$ and $B$ never jog the same distance.