Is this problem correct? Can we use '$\in$' quantifier between 2 sets? I got confused when saw the following problem: "$A, B, C: A \in B, B \in C, A \in C$. Do such sets exist?" Is it correct to use '$\in$' when we speak about sets, not elements? I'm new to the Set Theory so I don't know the subtleties. 

You have to define what are A,B,C before. In your case, they are sets so it is correct. Here's an example:
$A = \emptyset; B = \{ \emptyset \}; C = \mathcal{P}(B) $
$A \in B, B \in C, A \in C$
