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.
Three arrested after unrest in Plymouth
Man stabbed with a screw driver
They want to stop massive installation
"Shocking act of violence" say police
Photo released after Plymouth sex assault
Plymouth police are appealing for the public's help in tracing a person of interest following a sexual assault in the city on Friday.
Funding crisis causes closures
Five teenagers arrested after North Devon fire
Murder investigation underway
