Any lab measurement will be a terminating decimal, thus a rational number, because our instruments are not capable of infinite precision. You can, however, calculate an irrational number based on rational lab measurements, for example, by using the Pythagorean theorem or transcendental functions like sine, cosine, and tangent.
Ah, but the numerical values of the New System are mere approximations of the True Values of the Fundamental Constants. The True Values of the New System can only be expressed in terms of the Fundamental Constants and are thus independent of measurement.