Consider the following map $f$ from $\mathbb R^2$ to the torus $\mathbb T^2=\mathbb R^2/\mathbb Z^2$:
$$
 f(x,y) = (x, \sqrt2 y) \bmod \mathbb Z^2 .
$$
Its set of critical points equals its image and it is dense in the torus, so the $\epsilon$-neighborhood has full measure for any $\epsilon>0$.