Intuition for non-convergence of Cauchy sequence in $\mathbb{Q}$
Your Cauchy sequence is going somewhere, it's just not going to somewhere in $\mathbb{Q}$. At the end of your process you would be standing at $\pi$, so you would have left the rational numbers.
One can see even easier examples of this phenomenon: Consider the sequence $x_n = 1/n$. At each step of following that sequence you are in the interval $(0, 1)$, but at the end of the process you have "fallen out" of that interval.
The key point here is that, in general, if you have a Cauchy sequence in some set $A \subseteq \mathbb{R}$ the limit of that sequence doesn't have to be in $A$ anymore (but it can't run too far away from $A$: the limit will always be in the closure of $A$).
On one hand, it might be easier to think of the sequence $x_{n} = 1/2^{n}$ in $S = \mathbb{R} - \{0\}$. This is a Cauchy sequence but it has no limit within the set $S$, because the set has a "hole" at zero. Similarly, people describe $\mathbb{Q}$ as having "holes" almost everywhere.
However, I think this question deserves an entirely different approach. Instead of questioning the rational number system, you should question your own intuition here. Why should there be a limit?
The very idea that Cauchy sequences ought to converge comes from presupposing the structure of real numbers without even realizing it. It's very subtle.
I get this is kind of a weird question to ask, but I feel like it's hard for me to intuitively grasp what it means for a Cauchy sequence to not converge (since it feels like it is going somewhere).
Simply said, the rational number system neither conforms to usual intuition nor is it a useful system to model the outside world (Newtonian mechanics, general relativity, etc. all rely on using the real number system).
Your scenario is precisely why we are interested in the real number system, which does manage to do both, and the ultimate utility of the real numbers has to do with calculus. See this post.
There is a recurring theme in math where you think of something intuitive but not well-specified, and then you want to formulate that thing in more rigorous terms. That is, you think of something that ought to be possible, and then you rewrite that idea over and over until it is in the most precise possible form. (Sometimes the final product looks nothing like what you thought it would look initially.)
In your scenario, your intuition presupposes the real number system (but it is in the fuzzy stage as explained in the previous paragraph).
There is a list of axioms called the "ordered field axioms" that characterize fields (as you probably know, a field is a certain kind of arithmetic system) with an ordering relation. They characterize a wide class of fields, including the rationals.
However, it seems as if this list of axioms is "missing" something, because of your scenario. This is why we invented a new axiom called the completeness property, which is what your intuition is trying to articulate.
When you successfully formulate the axiom and add it your list of ordered field axioms, the list then uniquely characterizes the real number system (up to isomorphism).
What's important to see is that this convergence in the reals happens not because of your intuition, but because you constructed the real numbers specifically to conform to your intuition to begin with. And that intuition comes from your understanding of the outside world.
Suppose we were standing on the rational line at the point 3. Then we took a step to the point 3.1, then to 3.14, etc. (Cauchy sequence of decimal approximations of $\pi$). Suppose, also, that it takes us $\frac{1}{2^n}$ seconds to take the n'th step (so that we'll have "completed" the process after 1 second). Given that this sequence obviously doesn't converge in $\mathbb{Q}$, where would we be after continuing this process for 1 second?
Same place you would be if you jumped between $1$ and $2$ with each step taking $1/2^{n}$ seconds, or if you switched a light bulb on and off repeatedly for $1/2^{n}$ seconds. A person might think this contemplation of the supertask should give you an answer, but at the end of the day, there really is no answer.