Using data taken at 23 center-of-mass energies between 4.0 and 4.6 GeV with the BESIII detector at the BEPCII collider and with a total integrated luminosity of approximately $15$ fb$^{-1}$, the process $e^+e^-\to 2(p \bar{p})$ is studied for the first time. The Born cross sections for $e^+e^-\to 2(p \bar{p})$ are measured, and no significant structure is observed in the lineshape. The baryon pair ($pp$ and $\bar{p}\bar{p}$) invariant mass spectra are consistent with phase space, therefore no hexaquark or di-baryon state is found.
