$B$ mesons are the lightest mesons which can decay to various final states containing baryons. The measurement and comparison of exclusive branching fractions of baryonic $B$ decays, as well as studies on the dynamic of the decay, may allow better understanding of baryon production in $B$ decays and, more generally, hadron fragmentation into baryons. We present here a search for the decay of a $B$ meson in four baryons, $B^0\rightarrow p p \bar{p} \bar{p}$. From the observation of $(10.4 \pm 4.3)$ signal events, we provide the upper limit $\mathcal{B}(B^0\rightarrow p p \bar{p} \bar{p}) <2 \times 10^{-7}$ at $90\%$ CL. The data set consists of about $470$ million $B\bar{B}$ pairs collected with the BaBar detector at the SLAC National Accelerator Laboratory.