This is a trivially absurd argument for two reasons: First, it would be extremely straightforward to make a measuring device that spits out irrational numbers.

This is a trivially absurd argument for two reasons: First, it would be extremely straightforward to make a measuring device that spits out irrational numbers. Take the output, truncate at half the accuracy, and append an irrational to the output. Second, outputting an irrational number as a measurement does not imply that it's able to output any member of the complement of the rationals in the reals. You also conflate the computable numbers with the describable numbers -- but I'll give you the benefit of the doubt and assume you believe strong Church-Turing and aren't just committing an elementary error. There are ways in which set theoretic concerns apply to the real world, but they are few and far between, and this is not among them. You're essentially in line with people who attempt to use Goedel's proofs to make grandiose pronouncement about human thought. It. Does. Not. Apply.