I know there's a new html5 API called web storage, but it basically allows only up to 5MB of storage per domain, so how services like Deezer do to offer an offline mode ? It means that they have to store the musics on the computer, but it's all in the browser, so how do they do to bypass the 5MB of storage ?
I can't speak for Deezer, but there are several new APIs in development for HTML5 including the FileSystem and FileWriter API.
In use cases it gives examples of large storage requirements the API will be able to fulfil such as storage of large media files and assets.
It's still a working draft and not very widely adopted.