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 a lot of examples). For example, see the example just given by Lee Mosher.