the difficulty you point out is the constructivists' main bone of contention with ZF set theory. the problem lies with the power set axiom, if S is an infinite set, it is impossible to explicitly construct ALL the given subsets.
in a naive fashion one can say: "most real numbers are totally unknowable" (for to be described in any fashion, we have to give SOME adequate description, and surely there are only countably many of these, since we can only produce countably many "specification" schema in the finite time that is human history).
what happens is you get a version of downward lowenheim-skolem: a countable model of the real numbers (the ones we can actually define). it has been suggested by some that the power set axiom is "too wild", it allows for things to be automatically sets that we have no way of actually verifying WILL be sets (in an intuistic sense).
put another way: it is easier to decide logical questions in a first-order theory. but first-order theories have limitations: one of them is that "countability" is not absolute, but depends on which model you are using. because of this, most mathematics is actually done in second-order logic (but can be done in first-order logic with a few mental gyrations).
in actual point of fact, ZF set theory was originally formulated using urelements, but it turns out that "you don't get anything extra by including them". for example, there is little point in separating the "definable" real numbers from the "undefinable" real numbers, when constructing a proof of the mean-value theorem.
in other words: it is not necessary to have an EXPLICIT formula for defining a set, in order to know that something IS a set. we just need "other sets we can make A from" in order to say A is a set. i like to paraphrase this as:
"we have no freaking idea what sets are".
because of this, i myself prefer a more structuralist (categorical) view: it doesn't matter what sets ARE, as long as they behave themselves. i don't believe, in general, that we can form a "universal foundation" for mathematics, or even that such a thing is desirable: but rather, we should tailor our approach to the needs of the problems we are trying to solve.
in a more pointed answer to your question: the subsets of Re are sets, by DEFINITION (it follows from the axioms of ZF). the fact that we don't know "which" sets all of these are, is immaterial. if ask you to give me a SPECIFIC example of a urelement, as soon as you produce it: oops, now its not anymore.
let me be a bit more specific: assume we use the "cauchy-sequence of rational numbers" definition of the reals. the fact that i cannot list ALL the possible "convergence values" of all cauchy sequences, does not stop me from using their PROPERTIES (as limits of cauchy sequences) to deduce facts about them, as these properties can be "squeezed" from rational numbers, which no one has any quibble with (the one property of the reals that is troublesome, from a logical standpoint, is the "least-upper-bound property" (every non-empty set A has a supremum)). in most proofs that use the least-upper-bound property, there isn't any need to know what a real number x actually is (that is which cauchy sequence(s) of rationals its the limit of). most theorems in analysis (or calculus, in particular) are "non-constructive", we assert the existence of real numbers (such as the "c" in the mean-value theorem) without actually producing them. i find it preferable to assume we are working in a complete ordered archimedean field, without worrying overmuch about how we obtained it. realistically: how important is the "existence" of the real numbers? it's an IDEA. if we can communicate this idea effectively and consistently, that matters more than "what is its "true nature" ".