What is the product of the following ??? 4-methyl-3-hexanone with #"SH"-"CH"_2"CH"_2-"SH"#

Consider the possibility of a protecting group. You seem to be missing an acid catalyst, so I'll assume that you have an acidic solution, in which the ketone (pKa #~~ 20#) is protonated (by hydronium, for instance). That way, the thiol (pKa #~~ 10.6#) can act as a sufficient nucleophile.

(If there was no acid, essentially no reaction would occur.)

1. Not shown, but protonate the carbonyl with hydronium to catalyze the reaction.
2. The now-nucleophilic-enough thiol backside-attacks the electrophilic center.
3. Proton transfer to the leaving group, part 1
4. Proton transfer to the leaving group, part 2
5. Tetrahedral collapse; departure of protonated hydroxyl group
6. INTRAmolecular ring formation (faster than proton transfer, an INTERmolecular process)
7. Proton transfer to finish the reaction

Overall, this protecting group raises the pKa and steric hindrance of the originally-electrophilic carbon, allowing one to protect that carbon from further reactions. This is especially useful if there are multiple reactive carbons but you know you can target a specific one that is more electrophilic or more basic than the rest, for example.