Let me interpret your *constructive* terminology to mean *definable* with respect to some countable first order language, since although this may not be your original focus, there is something interesting to say about it.

One often hears it said that there must be reals that we
cannot define or describe, because there are only countably
many definitions, but uncountably many reals. And if one
considers only definitions over a fixed first-order
structure in a countable language, then this is correct for
the reason you describe. Nevertheless, there is another
very curious sense in which it can fail, which I would like
to explain.

An object $w$ in a first-order structure $M$ is *definable*
(without parameters) if there is a formula $\varphi(x)$
such that $w$ is the only object in $M$ satisfying
$\varphi(w)$.

For example, there are no definable reals at all in the
structure of the reals $\langle\mathbb{R},\lt\rangle$
as a pure order, because any two elements are
automorphic. In the ordered real field
$\langle\mathbb{R},+,\cdot,\lt,0,1\rangle$, we lose these
automorphisms and gain many definable reals, but only
countably many, because there are only countably many
definitions. More reals become definable with additional
structure
$\langle\mathbb{R},+,\cdot,\lt,0,1,exp,sin,\ldots,\rangle$,
but in a countable language, there are still only countably
many definable reals.

We can also extend the structure to higher orders, such as
second-order analysis or higher levels of the set-theoretic
hierarchy. For example, for a given infinite ordinal
$\theta$, one could look at the definable reals that exist
in the set-theoretic universe $\langle
V_\theta,{\in}\rangle$, which even for moderate $\theta$,
such as $\theta\gt\omega^2$, would include all of the usual
classical structure on the reals, as these are definable in
a comparatively small amount of set theory. But still,
since this language is countable, there would still be only
countably many definable reals.

Suppose we should go all the way? Let's consider the reals
that might be defined within set theory at all, without
cutting off the universe at $\theta$.

**Question.** Can there be a model $W$ of ZFC which thinks of
each of its reals that it is definable without parameters?

The curious answer is Yes! And this is the sense I mentioned at the begining in which the original inquiry can fail.

This property is already exhibited by what is
known as the minimal model of ZFC, the smallest
$L_\alpha\models ZFC$. Similar models can be built by taking definable
Skolem hulls inside any model with a definable Skolem
function. More generally, Jonas Reitz, David
Linetsky and I, building on results of Ali Enayat and Steve
Simpson, have proved:

**Theorem.** Every countable model $W$ of ZFC has a class
forcing extension $W[G]$, a model of ZFC, in which every object is definable
without parameters.

Thus, in the model $V=W[G]$, every object whatsoever
happens to be definable without parameters. In particular,
in this universe it happens that every particular real is
definable without parameters, even though the language is
countable.

The resolution of the resulting paradox is that the
property of "being definable" is not first-order
expressible. Although the model thinks that there are
uncountably many reals, each of which happens to uniquely
fulfill a definition, and it thinks that there are only
countably many definitions, it is nevertheless unable to
map those definitions onto the reals, since by Tarski's
theorem, there is no universal truth definition. So it is
unable to build the countable sequence of definable reals
against which we might diagonalize to produce a
non-definable real.