In constructive mathematics, we don't reject "all infinite constructions". The only axiom which we don't generally assume is the axiom which says "any statement is either true or not true". (Note that we also do not use the counterfactual axiom "there is a statement which is neither true nor false". In fact, we're just agnostic on some truth values.)
In constructive mathematics, there is a perfectly well-defined set of real numbers. The usual diagonalization proof that this set is not a countable set applies.
In constructive mathematics, there is a perfectly well-defined set of real numbers. The usual diagonalization proof that this set is not a countable set applies.