Yes, this is the canonical example of a non-trivial event with $0$ probability.
Probability is a measure, and it is quite common for non-empty sets to have $0$ measure (such sets might be dense and uncountable &c).
Conversely, an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability $1$. Note the qualifier almost! E.g., if you pick a random number in $[0;1]$, it will be almost surely an irrational, moreover, a transcendental number (because their complements - rationals and algebraic numbers - are countable and thus have zero measure). This does not mean that you cannot possibly pick $1/2$.
If you view probability as a subjective measure of likelihood that a certain event will occur, then, obviously, you cannot believe that one number in $[0;1]$ is more likely than another one; which means that each individual number has to be assigned probability of $0$.
If you view probability as the limit of frequency, then a random sequence in $[0;1]$ will probably contain no duplicates, so, as the number of trials goes to $\infty$, the number of successes (i.e., occurrences of the specific number) will be $0$ or $1$, so the probability will be $0$.