If the copyright of a written work is still in force, anyone who wants to make an audiobook from the work must get permission from the owner, who typically charges a fee.
However, if a written work is old, the copyright may have expired, in which case the work is in the public domain and there’s no need to ask permission or pay a fee. Anyone can make an audiobook from a text in the public domain. And that’s just what the volunteers at Librivox are doing!
Librivox has a catalog of thousands of fiction and non-fiction audiobooks made from public domain texts, and they give them away for free.
Some books are recorded by a single volunteer; others are collaborations between several volunteers who each take charge of whole chapters; others are dramatic collaborations between several volunteers who each take charge of one or more specific characters.
Librivox doesn’t have a “classic literature” section, but if you’re looking for classics, you can search for specific titles or authors, or browse within the following chronological categories:
Gutenberg.org, which supplies some of the texts used by Librivox volunteers, has a catalog of over 500 human-read audiobooks made from public-domain texts. Not all of them are in English. I’m not sure if they have any that Librivox doesn’t have, or whether all of their audiobooks are also Librivox audiobooks.