Abundance anomalies observed in globular cluster stars indicate pollution with material processed by hydrogen burning. Two main sources have been suggested: asymptotic giant branch (AGB) stars and massive stars rotating near the break-up limit (spin stars). We propose massive binaries as an alternative source. We compute the evolution of a 20 Msun star in a close binary and find that it sheds about 10 Msun. The ejecta are enriched in He, N, Na, and Al and depleted in C and O, similar to the abundance patterns observed in gobular cluster stars. In contrast to the fast, radiatively driven winds of massive stars, this material is typically ejected with low velocity. We expect that it remains inside the potential well of a globular cluster and becomes available for the formation or pollution of a second generation of stars. We estimate that the amount of processed low-velocity material ejected by massive binaries is greater than the contribution of AGB stars and spin stars combined, assuming that the majority of massive stars in a proto-globular cluster interact. If we take the possible contribution of intermediate mass stars in binaries into account and assume that the ejecta are diluted with an equal amount of unprocessed material, we find that this scenario can provide enough material to form a second generation of low-mass stars, which is as numerous as the first generation of low-mass stars. In this scenario there is no need to make commonly adopted assumptions, such as preferential loss of the first generation of stars, external pollution of the cluster, or an anomalous initial mass function. [Abridged]