Quick answer: In Barnstaple you can play board games at Board Game Night at 51° Coffee Roasters (games event). One more group is listed below; check their own page before you go.
Barnstaple's board game nights are small and seasonal. 51° Coffee Roasters opens its roastery at Chivenor for monthly winter board game nights (£10 with a drink), and TM Gaming on The Strand has 20 tables, with board game and roleplaying sessions usually at weekends. Exeter is the nearest town with weekly groups.
The 51° Coffee Roasters roastery at Chivenor, between Barnstaple and Braunton, opens up for board game nights through the winter. It's £10 with your first drink included, and the shelf runs from Catan and Splendor to Herd Mentality, Dobble and UNO. Book a space online first.
A tabletop gaming shop on The Strand with 20 tables and game nights every weekday from 5pm, mostly Warhammer and card games. Board game and roleplaying sessions usually run at weekends, with dates posted on its Discord, and any game can be played on any night.
Where
10 The Strand, Barnstaple, EX31 1EU
When
Shop Monday to Saturday 10am to 5pm; game nights from 5pm on weekdays.
Check the shop's own page before you go, as times can change.