The answer is no.
In fact, if $V$ is a Fano 3-fold containing $S$ as a hyperplane section, by the Lefschetz Hyperplane Theorem the restriction map $$r \colon \textrm{Pic}\,V \longrightarrow \textrm{Pic} \, S$$ is injective, and this is a constraint on the K3 surface $S$.
For instance, it is well-known that for all $g \geq 3$ there exists a polarized K3 surface $(S, \, L)$ of genus $g$ such that $\textrm{Pic} \, S$ is generated by $[L]$, see for instance
Picard group of a K3 surface generated by a curve
On the other hand, we know by the work of Iskovskih that smooth Fano threefolds with $b_2=1$ form a bounded family (they belong to exactly $17$ isomorphism classes).
Therefore, if $g$ is large enough, every smooth Fano $3$-fold $V$ of genus $g$ satisfies $b_2(V) \geq 2$, i.e. $\textrm{Pic}\, V $ has rank $\geq 2$. Since $\textrm{Pic} \, S$ has rank $1$, by the injectivity of $r$ it follows that $S$ cannot be obtained as a hyperplane section of a smooth Fano $3$-fold.
You can find a detailed discussion of this interesting problem in this paper by A. Beauville.