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.
Council spending less on social service agency staff
More than £3.6 million saved in two years
Parts of Central Park to get streetlights
Pilot scheme to launch by the winter
Teignbridge launches £300k 'Crisis Fund
Low-income families receive lifeline
More cliff falls expected at Shaldon
Tree surgeons working around clock
Seaton to bid for £2 million lottery funding
Go-ahead given to try and secure cash
Pine Martens thriving across Devon
Animals reproducing on Dartmoor and Exmoor
