The building is owned by South West Water and the museum's lease expired at the end of last year.
The trustees want to raise enough money to buy the building.
The museum contains exhibits from Exmouth's heritage, dating back hundreds of years, which would have to be given to other collections outside the town.
Chiefs plug coaching gap
England under 20s' attack specialist joins Sandy Park
Vehicle hits railway bridge
But services restored quickly
How empty churches could answer Torbay's prayers for new homes
'It's a win-win situation'
Anger as councillor in racism row is chosen as Torbay's deputy mayor
'The deputy mayor should be a Conservative but it shouldn’t be this one'
Devon trailblazer footballer gets blue plaque
John Leslie rejected for England because he was black
Devoncast - A pool protest, good news for tourism and a Dartmoor headache
Plus - a new mental health initiative and an MP goes into battle over parking
