The Market House Museum, in the small Georgian building on Market Street, is the town's local history collection. The displays cover Watchet's harbour history, the iron-ore trade with South Wales, the railway, and a particularly good section on the Saxon and Viking-age finds excavated locally — Watchet was a mint town under Alfred the Great and several of the coins struck here are on display.
It is staffed by volunteers and free to enter; opening hours are limited (usually afternoons in summer). Allow 45 minutes. The museum is the best single source on why this small harbour mattered far more historically than its current size suggests.