Can non-staff personnel in public institutions handle social security and medical insurance?

Of course. I am a supernumerary who is not prepared by a public institution, but the unit has to pay social security, medical insurance, pension and provident fund for me, all of which are deducted from my personal salary and paid by the unit.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.