The zoo said: "It was not an easy decision but it was the right decision."
Despite extensive in-house treatment supported by a range of experts it was concluded there was nothing more that could be done.
A spokesman said: "Her welfare was reduced and her quality of life deteriorating."
She had developed abscesses on her front feet in July 2008 and they had spread to all four. She also had arthritis.
She was put to sleep by the zoo's vet team early yesterday morning.
Senior keeper Jim Dicks, who has been working with Gay for 20 years, said: "It's like losing an old friend. We will all miss her — staff, volunteers, members, visitors.
"I hope people will remember being amazed to see elephants in the flesh at Paignton Zoo, because they are the most extraordinary animals."
His colleague Chris Bowers said: "Gay was a big, big part of my life and I'll never forget her."
Zoo director Simon Tonge announced her death to staff and then walked around the grounds to answer questions. He said many were half-expecting the news.
Curator of mammals Neil Bemment said: "Our vet team and animal staff did absolutely everything they could.
"We tried a range of treatments and consulted any number of experts.
"It became clear we had done everything we could for her and there was no option but to put her to sleep.
"It is very, very sad, but we know we have done the right thing for her.
"The welfare of the animal is paramount. Anyone who has ever cared for a pet or kept livestock has to face times like this.
"It was not an easy decision but it was the right decision.
"This is an upsetting and emotional day for the staff, volunteers, members and visitors to the zoo."
The large mammal house was closed for the day while the team conducted the grim task, and the following post-mortem examination.
Gay was given a general anaesthetic followed by a euthanasia agent which stopped the heart and breathing.
The zoo says the operation went smoothly.
After the post-mortem examination Gay will be removed and cremated off-site as Government regulations state this must be done and burial is not allowed.
The zoo says there are no plans to bring in more elephants and Gay's companion, Duchess, an African elephant, has her own health problems which the zoo is monitoring to make sure she maintains a good quality of life through the support of keepers and activities.
Dr Amy Plowman, head of research, said: "Duchess has been and will continue to be closely monitored to ensure she enjoys the best welfare possible."
Gay was born 1970 and arrived at Paignton Zoo in 1977 from Longleat Safari Park along with Duchess.
Gay had been previously with Fawcett's Circus in Ireland.
In 1999 the elephants moved from an old-fashioned enclosure into a large mammal house in the savannah habitat zone.
It was enclosure which suffered a fire in 2006 which killed three giraffes.
Gay and Duchess were saved by keepers, who were later commended for their bravery by the fire brigade.
The book of condolences is open for members and visitors to sign. Messages can be sent online to info@paigntonzoo.org.uk