This is a request for help, not an attempt to challenge anything.
Since $\pi$ is irrational, this tells me that it's impossible to express the distance around a circle in terms of the distance accross.
That boggles my mind, but maybe it should not.
I think crazy thoughts like: "this means that a the path of a circle around a unit lenth line segment has a non existant length".
Is there a way to accept that the number is irrational and not break from reason?
On
That $\pi$ is irrational does not mean that it's impossible to express the distance around a circle in terms of the distance across. It only means it can't be expressed as a ratio of integers.
BTW here you can read some proofs that $\pi$ is irrational.
It is quite easy to prove that $\sqrt{2}$ is irrational. It's even easier to prove that $\log_2 3$ is irrational. But it's hard to prove $\pi$ is irrational.
Here is a related question that may be of interest, where you will read about how to prove some numbers are irrational.
Some numbers are indeed impossible to express. Those are the numbers that are not definable. Easier to deal with, but still difficult, are numbers that are not computable.
However, it is not so difficult to provide a clear definition of the quotient of the circle's circumference by its diameter. We can let the matter stand there and simply name the quotient "pi", or we can do like Archimedes and define that quotient to be the limit of the regular polygon's perimeter to its diameter (using either choice of the inner or outer diameter) as the number of sides approaches infinity. We have the epsilon-and-delta definition of limits to help convince us that this is a valid definition.