Is it possible for a physical object to have a irrational length?

The set of irrational numbers densely fills the number line. Even assuming that quantum mechanics doesn't disable the preimse of your question, the probability that you will randomly pick an irrational number out of a hat of all numbers is roughly $1 - \frac{1}{\infty} \approx 1$.

So the question should be "is it possible to have an object with rational length?

Is it possible for a physical object to have an irrational length?

It's a bit of a philosophical question, but one could say this:

Just for fun, assume you have a perfect 45-degree right triangular piece of metal whose base and height is rational. Then it's hypotenuse is irrational because its length is the base times $\sqrt{2}$.

So it is possible to have a physical object of irrational length IF you can have a physical object of rational length.

ADDED: Suppose you cut a 45-degree right triangle out of a material based on a square atomic lattice, so the base and height each consist of $N$ atoms separated by $d$. Then the hypotenuse consists of $N$ atoms separated by $\sqrt{2}\times d$, so it's still not rational.

Suppose instead the material is based on a hexagonal lattice. Then all inter-atomic spacing would be $d$, but it would be impossible to cut a perfect 45-degree triangle out of it. In fact, the only triangle with rational sides you could cut out of it would be equilateral.

Suppose your infinitely precise caliper gives the answer $2.00000000000000\dots$ How would you know whether this is $2$ exactly, or if somewhere past the trillionth decimal it starts to deviate from $2$? How would you read your infinitely precise caliper?