One of the standard construction of the real numbers is the set of equivalence classes of rational Cauchy sequences. This definition is equivalent of the handwavey definition above (irrationals are the Cauchy sequences that don’t converge to a rational number and the reals are the rationals plus the irrationals.)

However almost any construction of the real numbers is challenging to give a simple explanation for. Even leading 19th century mathematicians didn’t truly understand the real numbers until Cantor.