In the normal distribution, the probability of taking one specific value is said to be zero (because there are an infinite number of values, so the probability of one specific value is infinitesimally small).
But then how can the probability of taking any value in a given range not equal zero? If all these values have probability=0 of being selected?
The term under which you can learn more about this is countable additivity. A probability measure is required to be countably additive, that is, the probability of the union of any countable family (that is, sequence) of pairwise disjoint events must be the sum of the individual probabilities. You’re forming the union of uncountably many events, and there’s no such requirement in that case.
A possible source of your sense of contradiction may be that you mistake “probability $0$” for “impossible”. An event can have probability $0$ and yet be possible. In fact, in case of a continuous distribution, all elementary events have probability $0$, and yet the probability that one of them will happen is $1$.