Depends on what you're measuring. To illustrate why that isn't a facetious response, consider the difference between 'measuring' pi, 'measuring' a meter and 'measuring' the mass of a proton. (Or, for that matter, the relative mass of three of something to one of it.)
It's worse than that: you also need an unambiguous way of determining whether the needle is overlapping a stripe.
Pick your method. It’s the ratio of a circle’s circumference to its diameter.
I think it's reasonable to say we can't truly measure pi, though.
And you can neither know nor measure a random real.
> We know pi in the sense of "a unique real number satisfying many useful properties".
We know it a lot better than that. We have efficient programs that output the numerical value of pi for as many digits as you want.
There's a bunch of real numbers we can identify that are far harder to make use of or approximate, and don't have easy exact description of their value.
Sure, but how would you compare those against a measurement?