There is

In a first version of this answer I claimed incorrectly (see Francesco's answer), that there are no examples inside smooth example as you ask for, because if $X$ is a simply connected threefold then $Pic(X) $ is free and for any surface $S$ inside $X$ the canonical bundle $K_S$ threefolds. In fact what I had in mind is the restriction of $K_X+S$, so $K_S$ well known fact that an Enriques surface cannot be $2$-torsion. Examples with $X$ singular exist: they are called Enriques-Fano threefolds and there are several papers on the subjecta hyperplane section of a smooth threefold.

FinallyI leave here the closing remark, concerning because it is perhaps useful.

Concerning the axample example of the Fermat quartic, notice that an involution of $\mathbb P^3$ has fixed points on any surface $S\subset \mathbb P^3$ so you cannot get a free involution this way. Blowing up does not improve things, because you replace a fixed point wih a curve all made of fixed points.

There is no smooth example as you ask for, because if $X$ is a simply connected threefold then $Pic(X) $ is free and for any surface $S$ inside $X$ the canonical bundle $K_S$ is the restriction of $K_X+S$, so $K_S$ cannot be $2$-torsion. Examples with $X$ singular exist: they are called Enriques-Fano threefolds and there are several papers on the subject.

Finally, concerning the axample of the Fermat quartic, notice that an involution of $\mathbb P^3$ has fixed points on any surface $S\subset \mathbb P^3$ so you cannot get a free involution this way. Blowing up does not improve things, because you replace a fixed point wih a curve all made of fixed points.