You can say it’s exactly 1 plus or minus some small epsilon and use the completeness of the reals to argue that we can always build a finer ruler and push the epsilon down further. You have a sequence (meters, decimeters, centimeters, millimeters, etc) where a_n is the resolution of measurement and 5*a_(n+1) determines your uncertainty.
However, at each finite n we are still dealing with discrete quantities, i.e. integers and rationals. Even algebraic irrationals like sqrt(2) are ultimately a limit, and in my view the physicality of this limit doesn’t follow from the physicality of each individual element in the sequence. (Worse, quantum mechanics strongly suggests the sequence itself is unphysical below the Planck scale. But that’s not actually relevant - the physicality of sqrt(2) ultimately assumes a stronger view about reality than the physicality of 2 or 1/2.)
> A professor sets up a challenge between a mathematics major and an engineering major
> They were both put in a room and at the other end was a $100 and a free A on a test. The experimenter said that every 30 seconds they could travel half the distance between themselves and the prize. The mathematician stormed off, calling it pointless. The engineer was still in. The mathematician said “Don’t you see? You’ll never get close enough to actually reach her.” The engineer replied, “So? I’ll be close enough for all practical purposes.”
While you nod your head OR wag your finger, you continuously pass by that arbitrary epsilon you set around your self-disappointment regarding the ineffability of the limit; yet, the square root of two is both well defined and exists in the universe despite our limits to our ability to measure it.
Thankfully, it exists in nature anyhow -- just find a right angle!
One could simply define it as the ratio of the average distance between neighboring fluoride atoms and the average distance of fluoride to xenon in xenon tetrafluoride.