The void, which goes under the road on Cage Lane in Thetford, is about 3m (10ft) long and 1.5m (5ft) wide, Norfolk County Council said.
A lamp-post that was precariously close to the hole has been removed by workmen and the road cordoned-off.
The BBC has been unable to get in touch with the museum, but a volunteer told the Eastern Daily Press at the scene it would be open as normal this Saturday.
Much of the original Dad's Army BBC TV series, based around the British Home Guard during World War Two, was shot in and around Thetford, using locations such as Thetford Forest.
Eyewitness Gavin Hodge said he noticed the hole at about 13:00 and said when he looked down it there was "quite a void" filled with masonry and rubble.
In a statement, Norfolk County Council said: "Part of it is under the pavement which is starting to give way but it also stretches under the road, which is intact but beginning to crack.
"Tomorrow the county council is planning to fill in the gap and make it safe. We're then planning to resurface the road on Monday with the hope that the road can re-open in the afternoon.
"There's no obvious reason why this gap has opened up. Anglian Water have investigated and have confirmed there are no leaks that would have contributed to it and there's no recent history of similar problems in the vicinity."
A sinkhole has opened up outside the Dad's Army Museum in Norfolk.