The fact that π is irrational has absolutely nothing to do with physically measuring circles or with infinity. We know the value of π exactly.
Perhaps you could share that exact value with the rest of humanity. And I mean the number value, not the nominal value.
π = 4 atan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + …)
Being irrational number, there are no finite number of digits (e.g. in decimal form or other base) to represent the pi value exactly. Nor can such value be expressed as ratio of integers.
Likewise, in your representation the ellipsis hides away the infinity.
If you can compute a number to any desired precision, then you know its exact value.
And this still has nothing to do with measurement of the physical world. We cannot measure anything exactly.