[bjl]

You can certainly pick a random real number from the unit interval.

[waqf]

Really? Go on, then, pick one and tell us what it is (or at least tell us what your procedure was).

[Houshalter]

One example in a finite space and time setting would be selecting a random point on the ground. Say by dropping a ball there or something. The exact coordinates it lands is a random real number. But the probability that it landed on those exact coordinates is exactly 0, hence a paradox.