A comprehensive answer to this question can be found on World Wide Words:
One [theory] points to William Buckley, a convict in the early days of European settlement in Australia, who escaped in 1803 from the short-lived penal settlement at Port Philip Bay (where Melbourne is today) and lived for 32 years with the Aborigines in southern Victoria, gaining the sobriquet of The Wild White Man, before giving himself up and being pardoned. The implication is that, like Buckley, you have no chance of success, it being assumed that you measure success by an escape to a part of Australia colonised by European immigrants. One problem is that Buckley died in 1856, whereas the expression doesn’t appear in print until 1895 (though that isn’t a conclusive objection, since phrases are often transmitted orally for years before they get written down and Buckley’s story became one of the most common anecdotes told about the early days of colonisation). The other possibility links it with the department store in Melbourne run by Messrs Buckley and Nunn, so that the expanded version, “there are just two chances, Buckley’s or none”, is a pun. However, that phrase isn’t recorded until 1953 and you need to have William Buckley’s exploits in mind before the pun achieves its full force.
You must take your choice. At this distance in time our chance of finding out which, if either, is right is roughly Buckley’s.