A practical application is for example in probability theory (or anywhere that deals with measures) such as this question:

If we generate a random real number from 0 to 1, what is the probability that it is rational?

Because we know that the continuum is so much larger in a sense than the set of rationals, we can answer this confidently and say the probability is zero, even though it is theoretically possible for us to get a rational number.

Statistics deals with similar scenarios quite frequently, and without it we wouldn't have the modern scientific method.