"Probability of selecting any real number [0, 1] is 0, but it's not impossible as inevitably some real number is chosen"

I often see this phrase used as an example of something with probability 0 being possible. It's not sitting well with me because I know I can't pick a random real number and I'm fairly certain it's not possible to build a machine that can either. If I understand correctly you have to add more restrictive boundaries to build such a machine even in theory. So that tells me that picking any random real number is both P=0 and impossible. Am I missing what makes this common phrase true?