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.)