Actually, it's just as possible to get a rope of the exact length pi/2 as it is to get a rope of the exact length 1. Which is to say, it is strictly speaking impossible (heisenberg's uncertainty principle and all that), but you can get an approximation as good as you want, being far more limited by physics than by the properties of Pi.
Now, if you wanted a rope of length 1+i, that would be a different problem.
Edit to add: whether two objects can be exactly Pi meters apart is a different question, one with no known answer - is space infinitely subdivisible, or is there actually a minimum possible length? We certainly can't measure that distance, but whether it exists or not may be a different question (some physicists will claim that if it can't be measured it doesn't exist in principle anyway, others are more open to this idea).
In current QM space itself is assumed to be infinitely subdivisible/continuous, but there are theories like Loop Quantum Gravity where it's not, and interpretations of QM like Copenhagen where unmeasured quantities don't have any definite value.
Now, if you wanted a rope of length 1+i, that would be a different problem.
Edit to add: whether two objects can be exactly Pi meters apart is a different question, one with no known answer - is space infinitely subdivisible, or is there actually a minimum possible length? We certainly can't measure that distance, but whether it exists or not may be a different question (some physicists will claim that if it can't be measured it doesn't exist in principle anyway, others are more open to this idea).
In current QM space itself is assumed to be infinitely subdivisible/continuous, but there are theories like Loop Quantum Gravity where it's not, and interpretations of QM like Copenhagen where unmeasured quantities don't have any definite value.