Good question. The same question can be asked of ordinary natural numbers, the negative numbers, infinitesimals as well as various orders of infinity.
The philosophy known as Mathematical Platonism argues that whereas we can see one bottle or three chairs we do not see the numbers one or three, so they contemplate a world outside of time and space - Platos Heaven, where these numbers exist. One could argue this is where we also place the more exotic notions of numbers.
Aristotle when he contemplated the infinite distinguished between actual infinity & potential infinity, and stated that actual infinities cannot exist but that we can imagine a potential infinity.
So, even in Platos Heaven, we cannot find the infinite.
But this doesn't appear to be correct, given what mathematicians have discovered, for they might say, the potential infinity is 1,2,3,...; and we can complete it at the first ordinal omega,- which I'll write as w; but then our friend, who has been watching this demonstration, and who rather mysteriously calls himself Socrates, says
"well, that isn't truly infinity - infinity is where you stop because you cannot go further, but I see here that you can, for I can continue with w+1, w+2, ..; and then one completes this one gets 2w! And so the pattern repeats, whenever the series is completed, we can see it again as the first term in a new series".
In mathematical parlance w is the smallest infinite ordinal, and its cardinality is equal to the smallest infinite cardinal; the next infinite cardinal, when is at an immense distance when looked at ordinally. We've already discussed the following series
ie 1,2,3,..., w,w+1,W+2,..,w2,w3,..w^2,..,w^3,..,w^w,..w^w^w,..,w^w^w^...
and this completes at what is called epsilon-0. This is still much smaller than the first uncountable ordinal, which is called omega-1. In fact to reach this using the kind of notation we've been doing is impossible, and there is an ordinal that measures exactly this called the Church-Kleene ordinal - omega-1-CK.
And this brings in a very useful understanding of what ordinals can be said to measure, at least in mathematical terms - that is proof-theoretic strength. The strength of arithemtic is epsilon-0; a natrual way to understand this is to imagine a proof of a theorem as a tree of propositions and note that trees are have ordinal type epsilon-0.
So at least we know that ordinals are useful to mathematician, in that they can be applied to something outside the theory of ordinals themselves. But this, though diverting, doesn't tackle your central complaint, which is are these ordinals obtained in the real world.
Now the paradox you mention is reminiscent of Zenos paradox of Achilles & the tortoise; and you mention the classical solution to this problem. Another solution is that there is physically a limit to subdivision - the atomic structure of matter, and perhaps of space.
However, as one begins to think about them, as you have, one begins to realise that the classical solution is only a solution, and perhaps not the only solution, and even perhaps the wrong solution - meaning not the best.
Socrates, again steps in here and says:
"Well, this is all very well, and very good; but this all about infinity in the guise of magnitude; and there are other senses in which this mysterious term the infinite can be used. What about the all? I see for example, in set theory one has the universal set, the set that contains everything, and so must contain every ingenious conception that you have of the infinite, and in fact those which you haven't conceived of yet".
Of course, Russell discovered that this notion leads directly to a paradox, which he solved by inventing type theory, and then one sees that again one gets a hierarchy, types, types of types, types of types of types...; so however one looks at it, infinity is not itself within mathematics, it always remains in potentia.
Socrates interrupts here, "you forget Spinoza, he had a beautiful system for describing everything, the all - the infinite of extension and thought, the reality of the real and the unreality of the unreal as minor modes in the major mode of the infinite of the Good and God; for whilst Plato showed us three worlds and Descarte reduced this to two, Spinoza asked - why two? - if there is one, it is sufficient unto itself, but where there is two, there will be more. It is possibly too beautiful to be true; possibly too beautiful to be understood".
Yes, and too beautiful to be untrue and too beautiful to explain...
