Not really. In all of our physical theories, curved paths are actual curves. So, (assuming circular orbits for a second) the ratio between the length of the Earth's orbit around the Sun and the distance between the Earth and the Sun is Pi - so, either the length of the path or the straight line distance must be an irrational number. While the actual orbit is elliptical instead of circular, the relation still holds.

Of course, we can only measure any quantity up to a finite precision. But the fact that we chose to express the measurement outcome as 3.14159 +- 0.00001 instead of expressing it as Pi +- 0.00001 is an arbitrary choice. If the theory predicts that some path has length equal exactly to 2.54, we are in the same situation - we can't confirm with infinite precision that the measurement is exactly 2.54, we'll still get something like 2.54 +- 0.00001, so it could very well be some irrational number in actual reality.

What does "actually exist" mean? Does Pi "actually exist"?

and, depending on how you define the rationals, -0.

https://en.wikipedia.org/wiki/Integer: “An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...)”

According to that definition, -0 isn’t an integer.

Combining that with https://en.wikipedia.org/wiki/Rational_number: “a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q”

means there’s no way to write -0 as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.