I think the thought behind the question might be the notion that if $\pi$ is the ratio of circumference to diameter, then why can't I measure the circumference of a circular object and its diameter, throw these numbers into a computer, and get $\pi$? If we could somehow precisely measure these two attributes, then perhaps we could get precisely $\pi$.
The first issue with this line of thinking is the idea that a perfect circle actually exists somewhere in nature. I don't believe this is the case -- the best one could ever do is construct/find some polygon with a very very large number of sides. Perhaps even an arbitrary number of sides.
The second issue with this line of thinking is the idea that it is possible to precisely measure something. Even with a perfect circle at the ready, the best one could ever do is measure its circumference and diameter to some arbitrary level of precision. One could only ever say that its circumference or diameter is at least this, and at most that.