  The aim of this paper is to construct Calabi-Yau 4-folds as crepant
resolutions of the quotients of a hyperk\"ahler 4-fold $X$ by a non symplectic
involution $\alpha$. We first compute the Hodge numbers of a Calabi-Yau
constructed in this way in a general setting and then we apply the results to
several specific examples of non symplectic involutions, producing Calabi-Yau
4-folds with different Hodge diamonds. Then we restrict ourselves to the case
where $X$ is the Hilbert scheme of two points on a K3 surface $S$ and the
involution $\alpha$ is induced by a non symplectic involution on the K3
surface. In this case we compare the Calabi-Yau 4-fold $Y_S$, which is the
crepant resolution of $X/\alpha$, with the Calabi-Yau 4-fold $Z_S$, constructed
from $S$ through the Borcea--Voisin construction. We give several explicit
geometrical examples of both these Calabi--Yau 4-folds describing maps related
to interesting linear systems as well as a rational $2:1$ map from $Z_S$ to
$Y_S$.
