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.
Five men remanded on Exmouth murder charge
They face trial next month
Giant-sized laughs at Northcott panto
Jack and the Beanstalk - REVIEW
Hundreds sign petition calling for village zebra crossing
Calls for Starcross crossing
Car parking fees to go up on Dartmoor
There will be a 20p increase for 24 hours.
Tens of millions to be invested in Devon's buses
Local leaders will decided how money is spent
