  A 2-torus manifold is a closed connected smooth n-manifold with a non-free
effective smooth $\mathbb{Z}^n_2$-action. In this paper, we prove that a
2-torus manifold is equivariantly formal if and only if the
$\mathbb{Z}^n_2$-action is locally standard and every face of its orbit space
(including the whole orbit space) is mod 2 acyclic. Our study is parallel to
the study of torus manifolds with vanishing odd-degree cohomology by M. Masuda
and T. Panov. As an application, we determine when such kind of 2-torus
manifolds can have regular m-involutions (i.e. involutions with only isolated
fixed points of the maximum possible number).
