Minimum number of sets of unreachable vertices for directed acyclic graph (DAG)
The problem is fixed-parameter tractable, where the parameter is the number of output sets.

Define an undirected graph $G'$ with an edge $(v,w)$ for each unreachable pair of vertices in the original graph.  Now you want to find a minimal edge clique cover of $G'$.  The minimal edge clique cover problem is fixed-parameter tractable, where the parameter is the number of cliques.

I don't know if there is a polynomial-time algorithm.  The minimal edge clique cover problem is NP-complete for general graphs.  I don't know if $G'$ has any special properties that makes the minimal edge clique cover problem easy on it.  (It's not necessarily chordal, unfortunately.)