The Internet Archive was founded in 1996 with the intent to store the world's digital content, such as web pages, music, moving images and e-books.
In October last year, the group announced that it had stored more than 10 petabytes - 10 million gigabytes - of information.
As well as the Wayback Machine, the group also runs other archiving schemes - such as TV Search and Borrow, a collection of US national news programmes that can be obtained, for free, on DVD.
The Internet Archive system was praised recently for allowing access to US government websites that were knocked offline thanks to the federal government shut-down last month.
Thanks to the fire, the scanning centre in San Francisco will need to be either heavily repaired or rebuilt, Mr Kahle said.
In the meantime, as well as donations, the group is seeking help in setting up scanning operations in different locations.