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.
City lose at home again
Kyle Wooton brace gives Stockport win over sorry Exeter
Calls for inclusion on Plymouth lord mayor role
It traditionally alternates between Conservative and Labour
Upgrades to Plymouth storm overflows to take place
Hoe bathing waters are priority to stop sewage discharges
Dartmoor finances facing biggest cut for years
'Do DEFRA think the Dartmoor pixie exists?'
Rare curlews and martens are leaving Dartmoor - for now
'We hope some of them choose to come back to the wonderful place that is Dartmoor'
Dartmoor phone mast gets the go-ahead
Better broadband and emergency cover promised
