Is there a rational way to conceptualize an irrational number? 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?
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.
Irrational numbers can and do correspond to length

Yes. An irrational number is a number that cannot be expressed as a ratio between two integers. It does exist as a length. For example, $\sqrt{2}$ is famously irrational, but it is extremely easy to show it as a length: just draw the diagonal of a square of side $1$.

Conceptualisation of irrational numbers

The whole rational vs irrational business started in ancient Greece. The Pythagoreans (the members of a secret group of mathematicians) believed that the whole universe could be described in terms of whole numbers: $1,2,3,4,...$
They were doing geometry, and they realised that some lengths were not integer multiples of other ones (more than $2$ times as long, but not quite $3$ times).

That's when they invented ratios: relationships between numbers. For example, $2$ is to $3$ just like $4$ is to $6$, in the sense that if you were to draw a rectangle with sides $2$ and $3$ and, next to it, one with sides $4$ and $6$, they would look the same (try it!). We express that relationship, by saying:
\begin{equation}
\frac{2}{3} = \frac{4}{6}
\end{equation}

Since they have faith in the integers, they believed that any length could be understood this way, two lengths coming from the simplest shapes: the circumference of the circle of diameter $1$ and the diagonal of the square of side $1$.

Now you know that it turns out that both $\pi$ and $\sqrt{2}$ are irrationals. So what does it mean?  It just means that it cannot be expressed as a ratio!

Check out this video for a brilliant exploration of all of this!