This is a privately owned beach but dogs are welcome on the beach (off lead), and on the Coast Path alongside the chalets (on lead). There is a large pay and display car park and from the beach there is a 2 mile walk one way to Minehead, or a 2 mile walk along the newly created Steam Coast Trail to Blue Anchor in the other direction. The holiday park at the beach has a cafe and kiosk for refreshments.
Yarn Market Hotel in the High Street.
The hotel has a cosy restaurant separate from the main restaurant in which dogs are welcome and offers soup, teas, coffees throughout the day and light meals at lunchtime. Dogs get offered a biscuit (with the permission of the owner!) and for people holidaying in the area a leaflet is available showing places where your dogs are allowed, such as pubs, tea rooms, attractions.
I have not stayed at the hotel as I am local but they have dog friendly rooms and guests also get a small dog gift pack when staying.
Dunster Castle and Gardens.
Although not able to go in the castle itself, dogs are welcome to walk on a lead throughout the castle grounds and gardens.
Even if not visiting the castle a walk can be taken around the old Deer Park fields, with a lovely river running alongside where dogs can enjoy a cooling splash in the Summer (or even the Winter if they feel so inclined!)