The digital service, which launched on Thursday, allows the public to buy and keep more than 60 years of BBC shows.
Shows will typically cost Â£1.89 an episode and can be downloaded through bbcstore.com and iPlayer.
Billed as the most comprehensive collection of BBC programmes published, it includes some banned programmes.
Among them, the 1976 Dennis Potter drama Brimstone and Treacle and the War Game, a 1965 depiction of the impact of a nuclear war in Britain.
Cult comedies such as Morecambe & Wise and Dad's Army will be available to buy digitally for the first time, as well as the legendary David Frost interviews.
In addition, popular dramas such as EastEnders and Holby City can be bought from the site just a day after transmission.
Exclusive content released to mark the launch includes Matt LeBlanc fronting a series of Top Gear clips in Top Gear: The Races, bringing together memorable moments from the show's racing archive, and a Best Of collection from the popular car show.
"It's a fabulous new service for the BBC that makes it easy to browse, buy and download some of your favourite BBC programmes ever... in much the same way as previously you owned them on DVD," said Marcus Arthur of BBC Worldwide.
Prices will be clearly marked on the site and are described as "broadly competitive", with series typically costing Â£6.99.
Mr Arthur told Newsbeat he hoped, over time, to offer "value added material" to shows, that can be purchased as part of the episode or as an optional extra.
At present the focus of the archive is on drama, comedy and factual entertainment - including a collection featuring some of Sir David Attenborough's most memorable documentaries - but in time, the BBC hopes to extend the service to include more content for children and more music programming.
Children's classic Muffin the Mule is currently the oldest piece of content available. There will also an "of its time" section to include content that might no longer be considered appropriate for broadcast.
The BBC said consumers will still be able to access its shows through subscription services such as Netflix, and the BBC iPlayer would continue to offer a free catch-up service for 30 days after shows have aired.
The BBC is making more than 7,000 hours of TV available to buy from its archives, including current shows and "lost gems", from the new BBC Store.