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.

We can get "exact" lengths in rational multiples of pi by cutting a rope/string to exactly one circumference of a circle of radius (length/2pi). Similarly you can get square roots by using the hypotenuse of the appropriate right triangle.

I don't think you can cut a String exactlyu to the circumference of a circle of a certain radius. It will always be longer or shorter.

Are you saying it's possible to cut a rope to length 3?

you could cut it to the point that it's effectively 3 from yours or mine point of view but mathematically its likely 3.0000000000000000000000000001 or something.

Precision on any cut in the real world will near impossible to do in absolute.

I think you need to clarify your definition of "exact" for this to be a productive line of discussion. Are you looking for a closed-form solution, a rational number, something that can be constructed with compass and straight edge, ...?