“Swept Away,” a world premiere musical at Berkeley Repertory Theatre, is at its heart a seafaring tale, set to the music of the Avett Brothers. The musical chronicles the lives of sailors manning whaling vessels in the late 19th century, when the reliance on whale oil to fuel lamps was fast declining. For its 90-minute running time, “Swept Away” is a riveting picture of the boat, which departs from New Bedford, Massachusetts, shadowed by ominous signs and superstitious forebodings.