I'm not quite sure what you mean by a set satisfying the Continuum Hypothesis: that's an axiom about the non-existence of cardinals between those of the natural numbers and the reals. Anyway, you can certainly construct well-ordered sets of the same cardinality as the reals: for example the set of all binary sequences (maps from N to {0,1}) with the lexicographic order (compare two sequences from the zero-th term up and order them by their values where they first differ).