The man, in his 50s, was believed to be wearing a gas mask and holding a gun when he stormed into the Ladbrokes branch on Crownhill Road in Plymouth, Devon, shortly before 7pm last night.
Brave customers disarmed the man, Devon and Cornwall Police said, and held him on the floor while others raised the alarm.
Police arrested the man, who was already unconscious, upon arrival, but he was declared dead a short time later.
The police watchdog, the Independent Police Complaints Commission (IPCC), has been informed of the incident.
A police spokesman said: "It quickly became clear the man was unresponsive and an ambulance was immediately requested by police who carried out CPR until they arrived.
"A few minutes later three paramedics arrived at the scene and continued to carry out further CPR. Around 20 minutes later the man was declared dead at the scene by the paramedics.
"This incident has been voluntarily referred to the Independent Police Complaints Commission."
Chief Inspector Ian Drummond-Smith said: "This is a very serious incident. We cannot speculate on what has actually taken place here today.
"A full investigation has now begun and while the IPCC carry out their investigation it is not appropriate to comment further."
The man has been identified but next of kin have not yet been informed, police said.
Ladbrokes declined to comment when contacted last night.
The Ladbrokes branch remained closed this morning with two uniformed police officers stood outside.
Several customers turned up at the grey two-storey building to try to place bets, but were turned away.
David Walker, 55, from West Park, Plymouth, said he walked in just five minutes after the incident happened.
"I went in about 6.55pm, to put a bet on the football," he said.
"When I went in there, they were on the floor, they were scuffling on the floor, I just thought it was just a fight or something.
"There were the two men, customers I think, on top of him, and the bloke on the floor. There was a bloke behind the counter and an old guy with glasses standing around.
"The bloke behind the counter told me they weren't taking any bets and I had to leave.
"I didn't see their faces, as they were holding the guy down at the back of the shop. I couldn't see any mask and there wasn't any physical movement, they were just holding him."
Mr Walker said he has only lived in the area for four to five weeks and did not know the staff or customers in the shop very well.
"I only came up to do a bet, I just assumed it was a fight over money or something until they said he'd tried to rob the place."
Ex-MoD policeman George, who did not want to give a surname, said he was not shocked to hear the news, claiming there was another incident when a man had to be thrown out not long ago.
"There was a fight a while back, I think it was a drunk who had come in there and he had to be removed by staff," he said.
The 79-year-old, who visits the branch daily, said he left at about 4pm, when there were still about five people in the shop.
He added: "Presumably there were quite a few people in there, there isn't normally at that time, that's probably why he thought he'd come and play silly buggers with a gas mask."
Ivor Green, 67, also from West Park, said he was not surprised at all that the customers had leapt to the defence of the staff.
"They would do that in a shop like this, it is our betting shop," he said.
"The staff become your friends, it's a little family, and it's no different to a member of your family being threatened. The guys in there would be protective of them.
"It's poetic justice really. I wish everyone that robs a betting shop would have a heart attack.
"That's absolutely the way it should unfold. He shouldn't have been doing that in the first place."
Mr Green, who uses the betting shop daily, but did not go yesterday because of the bad weather, added he hopes this incident is seen as a warning to others.
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