We prove that every real algebraic set V ⊂ R n with isolated singularities is homeomorphic to a set V ′ ⊂ R m that is Q-algebraic in the sense that V ′ is defined in R m by polynomial equations with rational coefficients. The homeomorphism φ : V → V ′ we construct is semialgebraic, preserves nonsingular points and restricts to a Nash diffeomorphism between the nonsingular loci. In addition, we can assume that V ′ has a codimension one subset of rational points. If m is sufficiently large, we can also assume that V ′ ⊂ R m is arbitrarily close to V ⊂ R n ⊂ R m , and φ extends to a semialgebraic homeomorphism from R m to R m . A first consequence of this result is a Q-version of the Nash-Tognoli theorem: Every compact smooth manifold admits a Q-algebraic model. Another consequence concerns the open problem of making Nash germs Q-algebraic: Every Nash set germ with an isolated singularity is semialgebraically equivalent to a Q-algebraic set germ.