In wireless distributed storage systems, storage nodes are connected by wireless channels, which are broadcast in nature. This paper exploits this unique feature to design an efficient repair mechanism, called broadcast repair, for wireless distributed storage systems in the presence of multiple-node failures. Due to the broadcast nature of wireless transmission, we advocate a new measure on repair performance called repair-transmission bandwidth. In contrast to repair bandwidth, which measures the average number of packets downloaded by a newcomer to replace a failed node, repair-transmission bandwidth measures the average number of packets transmitted by helper nodes per failed node. A fundamental study on the storage capacity of wireless distributed storage systems with broadcast repair is conducted by modeling the storage system as a multicast network and analyzing the minimum cut of the corresponding information flow graph. The fundamental tradeoff between storage efficiency and repair-transmission bandwidth is also obtained for functional repair. The performance of broadcast repair is compared both analytically and numerically with that of cooperative repair, the basic repair method for wired distributed storage systems with multiple-node failures. While cooperative repair is based on the idea of allowing newcomers to exchange packets, broadcast repair is based on the idea of allowing a helper to broadcast packets to all newcomers simultaneously. We show that broadcast repair outperforms cooperative repair, offering a better tradeoff between storage efficiency and repair-transmission bandwidth.