Reducing 3 SAT to 3 SET PACKING I'm trying to prove NP-hardness of 3 SET PACKING, which is a following problem: given a family of sets where each set contains 3 elements, decide whether the family contains k sets that are pairwise disjoint.

I tried reducing 3 SAT, but to now avail, but reduction of any other NP-complete problem would be helpful.
Your problem is known as 3-dimensional matching, or 3DM. It is one of the 21 problems proved to be NP-hard in Karp's original paper (number 17 on his list).