Among the following molecules/ ions, ${\mathrm{C}}_2^{2-},{\mathrm{N}}_2^{2-},{\mathrm{O}}_2^{2-},{\mathrm{O}}_2$. Which one is diamagnetic and has the shortest bond length?

According to MOT, molecules having unpaired electrons are diamagnetic.

As the bond order increases, the force of attractions increases and bond length decreases. 

$C_2^{2-} : \sigma 1s^2{\sigma }^{*}1s^2\sigma 2s^2{\sigma }^{*}2s^2(\mathrm{\pi }2{{\mathrm{p}}_{\mathrm{x}}}^1=\mathrm{\pi }2{{\mathrm{p}}_{\mathrm{y}}}^1)$

Hence $C_2^{2-}$ is diamagnetic and has shortest bond length.