Question: Can the 2-dimensional sphere $S^2$ be partitioned into four nonempty sets such that every circle in $S^2$ passes through just three of these four sets?

Here, "just three" means "exactly three", circle is in the ordinary sense, i.e. round circle (or say 1-sphere), not necessarily great circle. If "just three" means "at most three" as Alexandre Eremenko understood, then the answer is yes (there are many examples). For example, see the example just given by Lee Mosher.