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.
Police searching for missing teenager
He went missing nearly a week ago
Man charged with attempted murder in Exeter
Two people were taken to hospital with serious injuries
New Torbay dementia project will be 'world class'
'The work that is going on at Rowcroft is absolutely fantastic'
Brixham boil water notice mostly lifted
2,500 properties are still advised to boil water
Devon secures 27 new medical school places as Health Secretary visits Exeter
It's part of a national scheme to increase the number of medical students in England
Tackling Torquay’s anti-social behaviour problem
£20million in government cash earmarked for local projects
