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."

Many clubs are facing a major financial crisis and have not paid salaries for several months.
The footballers' union says the strike will go on until the players received what they are owed.
The strike involves players of more than 200 Argentine clubs.
The two matches postponed were Rosario Central v Godoy Cruz, in the city of Rosario, and San Lorenzo v Belgrano, in Buenos Aires.
The crisis began after senior officials at the Football Association (AFA) were implicated in a corruption scandal involving television broadcasting rights.
The season should have resumed a month ago after the summer recess.
But the crisis led the AFA to delay the start until this Friday.
To try to avoid the strike, the government of Mauricio Macri made a $22m (Â£18m) payment on Thursday to the Argentine Football Association.
The money was owed as compensation for the cancellation of a contract with the government, which held the broadcasting rights until this year.
But the players' union says the money is not enough.
They said most club presidents had spent and borrowed more money than they should have.
The death in 2014 of Julio Grondona, who was president of the Argentine Football Association for 35 years, triggered the crisis.
He was also vice-president of Fifa, the world football's governing body.
The United States authorities who investigated corruption at Fifa labelled Mr Grondona "the number one conspirator".
But analysts in Argentina say he guaranteed the functionality of Argentine football.
Many believe, however, that his absence will allow a perhaps tortuous but necessary clean-up of the game, says the BBC's Daniel Pardo in Buenos Aires.

Seventh-placed Vikings went into the break 8-0 ahead thanks to two tries from Corey Thompson.
Liam Farrell's converted score brought Warriors back to within two against a side who defended stoutly despite at one stage having two in the sin-bin.
Wigan turned down a late chance for goal, but the visitors held out.
Shaun Wane's side need only one point to confirm a top-four finish, but decided against trying to draw the match when they won a 79th-minute penalty.
They decided to run it and, after it came to nothing, Widnes saw the clock down.
Wigan: S Tomkins, Charnley, Gelling, Sarginson, Tierney, Williams, Shorrocks, Sutton, Powell, Flower, Farrell, Bateman, Isa.
Replacements: Tautai, Gildart, Nuuausala, Bretherton.
Widnes: Hanbury, Thompson, Dean, Runciman, Marsh, Mellor, Brown, Burke, White, Buchanan, Whitley, Houston, Cahill.
Replacements: J Chapelhow, Dudson, Heremaia, Farrell.
Referee: Phil Bentham (RFL)
A sinkhole has opened up outside the Dad's Army Museum in Norfolk.
[NEXT_CONCEPT]
The Argentine Football Association has postponed two top division matches that were due to take place on Friday following the collapse of negotiations to call off a players' strike.
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Widnes Vikings produced a superb defensive display to beat Wigan and leave Warriors still waiting to confirm their Super League play-off place.