I believe you're mistaken. A Planck length is the minimum length we can extrapolate down before physics gets weird, but that doesn't mean it is the smallest possible length anything can be.
And an irrational number does exist as a discrete unit, it simply cannot be described as a fraction. Case in point, if you could create a spherical particle that was exactly 1 Planck length across, it would have a circumference of exactly π Planck lengths.
By your logic, such a theoretical particle could not exist because the circumference includes an irrational number in the size of the body.