Chondrite meteorites are the building blocks of the solar nebula, out of which our Solar System formed. They are a mixture of silicate and oxide objects (chondrules and refractory inclusions) that experienced very high temperatures, set in a matrix that remained cold. Their prevalence suggests that they formed through a very general process, closely related to stellar and planet formation. However the nature and properties of the responsible mechanism have remained unclear. The evidence for a hot solar nebula provided by this material seems at odds with astrophysical observations of forming stars. These indicate that the typical temperatures of protostellar disks are too low to melt and vapourise silicate minerals at the radial distances sampled by chondrule-bearing meteorites. Here, we show that processing of precursors in a protostellar outflow at radial distances of about 1 – 3 AU can heat them to their melting points and explain their basic properties, while retaining association with the colder matrix.