"I'm so moved, I think it's so fantastic that we get to see the history of Abba," said 46-year-old Swede Henrik Ahlen, who lives in London but came to Stockholm to be one of the first to tour the new museum.
"I was eight years old when they won the Eurovision Song Contest (in 1974) and they have always been a part of me."
Like many of the first visitors, most of whom were in their forties and all of whom were taking pictures, Ahlen had tears in his eyes as he looked around.
The museum features a host of exhibits including the glitzy costumes worn by the group, which has sold more than 378 million albums worldwide.
A 31-year-old Argentinian woman named Celeste, who said her grandmother raised her on Abba music, said she could "spend the whole day in the costume room."
"I've already been to Sweden eight times and every time it was Abba-related," she said, adding that she learned Swedish because of the band and that she had had five Abba costumes sewn up for herself.
She was joined by fellow Argentinian Mariano who said that Abba like no other musicians for him.
"I flew 22 hours just to be here. I have been an Abba fan for 40 years, it started when I was a child and has continued until today," he told The Local.
The quartet dominated the 1970s disco scene with their costumes, kitsch dance routines and catchy melodies such as "Voulez Vous", "Dancing Queen" and "Waterloo", the song that won the 1974 Eurovision Song Contest and thrust the band into the international spotlight.
They last performed on stage together in 1982 and split a year later, and have vowed they will never reunite to sing together again.
"There is simply no motivation to regroup. Money is not a factor and we would like people to remember us as we were," band member Björn Ulvaeus, 68, said in a 2008 interview.
On Monday, Ulvaeus, Anni-Frid (Frida) Lyngstad and Benny Andersson attended a VIP event at the museum. Agnetha Fältskog was promoting her latest solo album in London and did not attend.
The state-of-the-art museum, located on Stockholm's leafy island of Djurgården, allows visitors to get up close and personal with the band in interactive displays.
In one room, fans who have dreamt of becoming the fifth member of the band will be able to appear on stage with the quartet and record a song with them thanks to a computer simulation.
In another room dedicated to the song "Ring, Ring", a 1970s telephone will be on display. Only four people know the phone number: the four Abba members, who may occasionally call to speak live with museum visitors.
Other rooms feature childhood photos, the band's costumes and instruments, gold records, replicas of their recording studio and dressing rooms, and their stylist's worktable.
Visitors get the band's inside story told "with humour and warmth. They'll get close to the truth," Ulvaeus, who was married to Fältskog, told reporters on Monday.
Andersson and Lyngstad were also married.
"We also talk about daily life, life with the children, our break-up, the crises, things we haven't talked much about, the divorces. We've gone beyond the happy image that we presented," he told AFP in an interview.
Opening-day visitors gushed with excitement.
"We like everything in the museum but the interactive things are truly great: being on stage, singing in the studio and then you can download it all," said 49-year-old Claudia who came from Germany for the occasion.
"For me it's the only way to see them since I wasn't born when they split," said Anna Wagner, 29, from Dusseldorf.
Yet the entrance fee may deter some. At 23 euros, or $30 a ticket for anyone over the age of eight, a family of four with two children will have to dish out 91 euros for a few hours of fun.
But that hasn't stopped the most die-hard fans, who have booked up most of the available tickets online for the first few weeks, the lion's share of them from abroad, according to museum officials.
The museum says it expects to attract a quarter of a million visitors in 2013.
Stockholm Visitors Board spokeswoman Ann-Charlotte Jönsson said the museum "will be an international attraction."