The former Havant & Waterlooville boss takes over at Priory Lane and will straight away start building his squad for next season.
He signs a two-year deal and replaces Mark McGhee, who could only win one match since replacing Jamie Howell in February.
Bradbury left Havant at the end of the season after the club failed in their bid to remain in the National League.
He spent six years at the club, twice winning promotion - including the National League South title.
Bradbury said: “Right across Non-League football, Eastbourne Borough carry huge respect.
"I always enjoyed my contests against the Sports, and just as importantly the dealings with the great bunch of people who run the club and support the club.
"I believe that, knowing the National League as well as National League South pretty much inside out, I can bring my experience to take the Sports forward – and to the next level. I cannot wait to get started.”
Chairman John Bonar said: “The appointment follows a very thorough selection procedure, in which a strong list of applicants for the post was whittled down to a target list of about half a dozen candidates.
"All of those candidates impressed the Board, and indeed any of them could have done a fine job for Eastbourne Borough.”