I arrived at this question through my frustration that, despite my master degree, I could not come up with the proof of pi's irrationality just like that. So I studied it and wondered, why was this not on the list of things we learnt at university.
The question is different for an active professional mathematician, a high school math teacher or someone who is otherwise orbiting in our society with a mathematical education in the bag.
I would like to be able and answer questions by non-educated but interested people and picture them a background for the facts. The irrationality of Pi is a likely candidate for Christmas Eve questions, as is the infinite number of primes, or even Gödel's theorem. I studied that one too and it made a lasting impression on me.
In terms of relevance for society, an accomplished mathematician these days should be there to point out flaws in logic and bring an enhanced intuition of statistics to the public domain. Newspapers are full of "significant research results" and their interpretations. People are developing certain common knowledge while mostly remaining ignorant about the statistical aspects of that knowledge, as Daniel Kahneman has pointed out.
