By repeatedly throwing a needle on a striped pattern: [1]. Obviously, you will need an infinite number of throws for an infinitely precise measurement of pi.
That would affect only a few borderline trials and would average out with subsequent throws. It would be much more worrisome that the length of a needle or the width of a stripe is not infinitely precise, that would consistently affect all the trials.
It's worse than that: you also need an unambiguous way of determining whether the needle is overlapping a stripe.