The irrationality of pi is not exactly what the problem is. There are many rational numbers that cannot be expressed exactly in floating-point formats either, like 1/3 or 0.1.
Edit: sicrane brought me up to speed with your logic regarding 0.1. It's all obvious now that what does not possess infinite non-zero placeholders right of the decimal point in base 10 may have infinite non-zero placeholders right of the binary point in base 2. Thanks!
I vote for the next version of IEEE fp to be in base pi. Problem solved! I'm kidding.