Examples of the events for which we cannot assign "meaningful" probabilities

Quote from the book I'm reading:

Any collection of possible outcomes, including the sample space $\Omega$ and its complement, the empty set $\emptyset$, may qualify as an event. Strictly speaking, however, some sets have to be excluded. In particular, when dealing with probabilistic models involving an uncountably infinite sample space; there are certain unusual subsets for which one cannot associate meaningful probabilities.

Question 1

• What is meant by "meaningful" probabilities?

Question 2

• Can you provide an example in which we cannot assign meaningful probabilities to the events of the sample space?

from Hot Weekly Questions - Mathematics Stack Exchange