Why is the carbon-carbon bond in hexachloroethane shorter than in ethane?
One reasoning would be applying Bent's rule: 


In ethane the $\ce{C-H}$ bonds are approximately composed from $\ce{61\% sp^{3.3} C}$ and $\ce{39\% s H}$ orbitals.
That results that the $\ce{C-C}$ bond is composed from $\ce{sp^{2.3} C}$ orbitals.   
In hexachloroethane the $\ce{C-Cl}$ bonds are composed from $\ce{48\% sp^{3.7} C}$ and $\ce{52\% sp^{4.8} Cl}$ orbitals.
That results that the $\ce{C-C}$ bond is composed from $\ce{sp^{1.8} C}$ orbitals.


Higher $\ce{s}$-character in bonds usually means that the optimal overlap of these orbitals is at shorter inter-nuclear distances. Since the $\ce{s}$-character in the $\ce{C-C}$ bond in hexachloroethane is much higher than in ethane, the bond length is expected to be shorter.

(Values obtained from an Natural Bond Orbital Analysis on the DF-BP86/def2-SVP level of theory.)