Never mind pi for a bit. Suppose someone asks you to calculate say, the area or perimeter of a square formed by putting together four tiles, square in shape, each measuring one unit on a side. Simple enough for you?
If you answered: "Yes," then you are one ahead of me, roughly speaking anyway. It certainly is not simple enough for me.
Did you by any chance get the answer: four? Are you sure? Four point what? It is not enough to say four point nothing, if I ask you what the millionth digit behind the decimal point would be, assuming anyone needed to know that, you either would have to accept that that digit is necessarily a zero, or alternatively that the answer is not in fact four. And you would have to prove that your measurement really did determine that digit to be zero.
The same applies to any calculation whatsoever to which the answer is a single, definite, numeric value, whether an integer or not.
If the problem is abstract and the answer is correspondingly abstract, then that does not matter. We can speak of precise answers in practical fact (such as indeed four, or perhaps 3.3 recurring) where we know, or can trivially determine every single digit of the infinite decimal expansion, or answers in principle such as root 2 or pi, where the universe is not large enough for us to calculate an arbitrarily remote digit.
Let's get back to the area of our four tiles. We are already in a state of sin in that we accepted that each tile was square and measured one unit on a side. Even if hypothetically that was somehow meaningful at the start, it instantly stopped being meaningful thereafter. The tile is made of atoms. Atoms move. Atoms do not permit the measurement of macroscopic objects to a precision finer than a small fraction of the dimensions of an atom. Correspondingly no single number can describe any such parameter of a real-world macroscopic object.
Why then do we use numbers (as a general rule pretty satisfactorily) in describing everyday parameters of everyday objects?
This is a point open to a great deal of quibbling, but roughly speaking (much as we use such numbers roughly) you could say that it is because for most purposes the real-world objects differ from ideal objects only in parameters that we can afford to ignore and cannot in practice afford not to ignore.
Conversely, if we do ignore the negligible parameters, we generally can get thoroughly satisfactory correspondence to simple mathematical operations, such as the calculations of counts, areas, perimeters, masses, and so on. This I assume is more or less what you meant when you spoke of "a real relationship in the real world".
Such a relationship is only arguably real, and in fact is of limited precision, but generally it is close enough for jazz.
If on the other hand you are referring to a notionally infinitely precise calculation such as the hundredth digit of pi as applied to the measurement of a real-life approximation to a circle, then your calculation no longer has a meaningful relation to any physical measurement.
As the result of a calculation it is in fact no more than the implication, the inferred result of the implications, of certain assumptions. (If you prefer you might say that ultimately your results follow from your axioms.)
There is nothing wrong with that, but it does demand a healthy helping of common sense in applying such ideas in practice.
If that fails to clarify matters for you, please let us know, and we can have another go.
