Barnard Castle is a market town on the north bank of the River Tees, in County Durham, England. The town is named after and built around …Barnard Castle is a market town on the north bank of the River Tees, in County Durham, England. The town is named after and built around a medieval castle ruin. The town's Bowes Museum has an 18th-century Silver Swan automaton exhibit and paintings by Goya and El Greco.