Dunsborough is a coastal town in the South West of Western Australia, 254 kilometres (158 mi) south of Perth on the shores of Geographe Bay.he modern town is named after the nearby Dunn Bay, which was named after Captain Richard Dalling Dunn under whom Governor James Stirling had served in the Hibernia and the Armide in 1810-11. Land for a townsite was set aside here in the late 1830s, and there is a recorded whale fishery at Dunsbro in 1850.