There are many ways to show that the field of real numbers "exists" in the sense that a field with the desired properties can be defined from the field of rational numbers. The most popular ones seems to be Dedekind cuts (see e.g. "Set theory for guided independent study" by Derek Goldrei), and metric space completion (see e.g. "Foundations of modern analysis" by Avner Friedman, for a proof that metric spaces can be "completed"). I very much doubt that there could be a way that's much simpler than those. I like the metric space completion method, because it has other uses that are significant to physicists, so it's less likely to be a waste of time to learn it.
These constructions clearly do use only a finite number of words. What your professor was talking about is how a field that's defined using one of those methods contains members that can't be specified in a finite number of words. The argument is simple: Every real number has an infinite decimal expansion (possibly ending with infinitely many zeroes). If we can specify that decimal expansion in a finite number of words, we can write a computer program that generates all those decimals, one at a time. The program may never finish its run, but the program itself is a representation of that number. The problem is that there's only a countable number of computer programs. This implies that a number that can be generated this way belongs to a countable subset of ℝ, but ℝ is uncountable, so that subset can't be equal to ℝ.