Something I posted on reddit answers this question quite well, I think:
"Rational" and "irrational" are properties of numbers. Quantities with units aren't numbers, so they're neither rational nor irrational. A quantity with units is the product of a number and something else (the unit) that isn't a number.
By choosing the unit you use to express a quantity, you can arrange for the numeric part of the quantity to be pretty much any number you want (though switching units won't let you change its sign or direction). In particular, it can be rational or irrational. And choices of units are a human convention, so it wouldn't make any sense to extend the idea of rationality or irrationality to the quantity itself.
You can use a natural unit system, where certain physical quantities are represented by pure numbers. For example, if you use the same units to measure time and space, $c = 1$. In such a unit system, it does make sense to say the speed of light is rational, but that's kind of a special case. That reasoning doesn't really work with other physical quantities. And you really do have to be using natural units. (Technically, you could make a natural unit system where $c = \pi$, but it would have very complicated and perhaps even inconsistent behavior under Lorentz transforms, so nobody does that.)
By the way, empirical measurements always have some uncertainty associated with them, so they're not really numbers either and are also neither rational nor irrational. A measurement is probably better thought of as a range (or better yet, a probability distribution) which will necessarily include both rational and irrational numbers.