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 injured in Plymouth street fight
Police are appealing for witnesses
Pedestrian dies in Brixham crash
Articulated lorry involved
Body found at Sandy Bay
It's after two men went missing on Christmas Day
Raw sewage pours into landmark Devon river
Storm overflows gush into waterway for 40 per cent of year
Work underway to return shingle to East Devon beach
Bulldozer in action for Storm Goretti clean up
