I think the precise definition of what exactly is physically meaningful is more interesting a bit, because pi is just as 'real' as 1 in some sense. You can't any more precisely measure 1m than pi meters. A perfect square doesn't exist any more or less precisely than a perfect circle.

Basically, the universe can just as well be approximated in two different ways. One, it's all straight lines, and anything that looks like a circle/curve is actually a piece of a veeery many sided polygon if you look closely enough at it. Two, it's all curves of some curvature, and anything that looks like a straight line is actually a segment of a veeery large circle. The first perspective corresponds to taking the rational numbers as physical. The second one corresponds to taking pi as physical (and then it's unclear whether e is also physical, or which other transcendental numbers are, but 1 is definitely un-physical then).

Probably the constructive framework is the best way to express this concept, just starting from whatever constant we chose to define as the unit.

Note that π, like practically any constant or number normally found in mathemathics is constructive and contains a finite amount of information. What the author consider unphysical are the non-constructive reals: unfathomable numbers that can't be defined or written in any finite amount of words or paper. It's hard to argue they play some role in our physical world: it would basically amount to metaphysics.

However, even if rarely used outside of proofs, they are the vast majority of the reals, in fact there are only countably infinite constructive/computable numbers.