Shirley Taylor said the bees took residence in her Nissan car parked outside her front door in Watt Street, on Sunday.
Chairman of the Beverley Beekeepers' Association, Chris Coulson, has been trying to lure the swarm out of the vehicle since.
He said it was not clear what had attracted the bees to the vehicle and it could take a while to clear them.
Mrs Taylor said the bees had arrived on Sunday when she received a message from a neighbour warning her to be careful of her car when she got home.
"I just went indoors and closed all the windows and the neighbours did as well," she said.
"It has been a bit of a hair-raising experience.
"There were thousands all over the place and I have been stung and my daughter and granddaughter also got stung.
"It is like Nightmare on Watt Street."
Mrs Taylor said: "I did ask 'why pick on my car?' but my husband, who is a bit of a joker, said it was because of all the Bee Gees CDs in the car."
Mr Coulson said bees usually swarm when their hives get too large and a colony breaks away to form a new one.
However, he said he had never seen such a large swarm descend on a car before.
"These cars have all kind of recesses and the bees seem to have gone into every one they could find.
"We are trying to make them fly. In the box on top of the car we have young bees, some eggs and things like that and the bees in the car will hopefully try to cover those to make sure they don't die."
He said he hoped they could remove the bees and find them a "more acceptable" new home.
A swarm of up to 20,000 bees has taken over a car in Hull.