Say you have shown on paper that some algorithm is easy to compute (polynomial time with respect to input size $|X|$. It is clear to me that for a practical machine (essentially any current computer available to me, in the world), solution to a given problem with sufficiently large input size, is not only hard to compute the answer to, but impossible! The impossibility would come from the computer not being able to store any "faithful" representation of the output.
What axioms address this concern, or is it just neglected to be addressed across the board?
It seems like a good idea to consider the theoretical machine to have an infinite number of possible symbols (versus the 256 possible bytes), and an infinite amount of memory (versus the finite memory on a real machine). Where an algorithm is only considered possibly easy in time complexity if it has a finite representation on said infinite machine. Therefore, for some problems that are theoretically, actually easy to compute; they may be impossible to represent faithfully (in general) and compute on my home computer.
This is a highly confusing aspect of time complexity for me, and I am currently not comfortable with ignoring it. Take any general problem, such as sorting a list. If I can't even fit the list on my machine, then what!? Another example: multiplying two big integers in general "is easy" assumes that I can fit the every growing integer inputs on a machine!
Please guide me to the proper discussion for this, or convince me that my worries are not neccessary.