Choosing something of $0$ probability

What you're looking at here is the difference between impossible and probability zero. An impossible event is one which literally cannot happen; for instance, choosing a random number between $0$ and $1$ and getting a result of $5$. These events aren't even conceivable within the universe of discussion.

A probability $0$ event is one which is conceivable within the universe we're discussing, but which has no positive likelihood of occuring. Your example -- choosing a random number in $[0,1]$ and getting $0.5$ -- is a great one. Another good example is the event that a coin is flipped, then flipped again, then again... and so on forever, and always ends up heads. It could happen... but in (literally) all probability, it won't.

Remember that mathematics deals with an idealized model of the world. In reality you cannot observe an event that has an infinite (let alone uncountable) number of possible outcomes; even if the underlying physics of your random process is truly continuous, you cannot measure the $0.5$ outcome exactly. You might only be able to say for sure that the outcome was between, say, $0.499999$ and $0.500001$. And the event that $0.499999 < X < 0.500001$ where $X$ is uniformly distributed over $(0,1)$ has a non-zero probability.

In the idealized mathematical model, on the other hand, probability measure of a continuous random variable is a matter of integral calculus. The probability of any one exact observation is zero (the result of multiplying a zero-width interval by the pdf at that point), but the total probability over all the observations in a range is the integral of the pdf over that range, which is non-zero.