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.

The New Zealander made only one unenforced switch, bringing in Rhys Webb for Gareth Davies at scrum-half.
Sam Warburton, Alun Wyn Jones and Alex Cuthbert miss out with injuries.
"We wanted to give the players a chance to sort of put behind us a disappointing first half from last week," said Gatland.
Flanker Justin Tipuric, second row Luke Charteris and Hallam Amos are drafted in to the team in place of the injured players.
Dan Lydiate will captain the side in the absence of Warburton and regular stand-in Jones.
Gatland said he had shown faith in players who had performed well in earlier matches, a fact he acknowledged was hard on hooker Ken Owens who makes his fourth appearance on the bench in this Six Nations championship.
"He's unlucky, really unlucky," he added.
"We felt that [hooker] Scott Baldwin has gone pretty well throughout the campaign.
"Ken has been brilliant for us coming off the bench and I know that's a tag he doesn't want to keep, but he is unlucky not to get a start."
Scarlets hooker Owens had made 38 appearances for Wales, but started in only eight of those games.
Wales go into the match against Italy knowing a win will secure second place in the championship for the first time since it was expended to include six countries.
Lydiate will lead Wales for the first time in a Test match, having previously led them in a midweek match against EP Kings on the summer tour of South Africa in 2014.
He said it was a "personal honour for me and my family" and added his captaincy style was unsophisticated.
"I wouldn't ask someone to do something that I wouldn't do myself, so I'll throw myself in front of a bus so that's what I expect everyone else to do," he said.
"There are plenty of leaders and a wealth of experience in the team."
A swarm of up to 20,000 bees has taken over a car in Hull.
[NEXT_CONCEPT]
Wales coach Warren Gatland resisted making more changes to his team against Italy to give his men a chance to make up for their poor start at Twickenham.