Motivated by recent models proposing a hidden sector with $\sim$GeV scale force carriers, we present a search for a narrow dilepton resonance in 4 lepton final states using $536fb^{-1}$ collected by the BaBar detector. We search for the reaction, $e^+e^-\to W^\prime W^\prime\to(l^+l^-)(l^{\prime+}l^{\prime-})$, where the leptons carry the full 4-momentum and the two dilepton pair invariant masses are equal. We do not observe a significant signal and we set 90% upper limits of $\sigma(e^+e^-\to W^\prime W^\prime\to(e^+e^-)(e^+e^-))<(15-70) ab$, $\sigma(e^+e^-\to W^\prime W^\prime\to(e^+e^-)(\mu^+\mu^-))<(15-40) ab$, and $\sigma(e^+e^-\to W^\prime W^\prime\to(\mu^+\mu^-)(\mu^+\mu^-))<(11-17) ab$ in the $W^\prime$ mass range between 0.24 and 5.3GeV. Under the assumption that the $W^\prime$ coupling to electrons and muons is the same, we obtain a combined upper limit of $\sigma(e^+e^-\to W^\prime W^\prime\to(l^+l^-)(l^{\prime+}l^{\prime-}))<(25-60) ab$. Using these limits, we constrain the product of the SM-dark sector mixing and the dark coupling constant in the case of a non-Abelian Higgsed dark sector. Show Partial Abstract