Some of the transcriptions of records can be accessed at FamilySearch.org (free with subscription). No, not just those over 65, but must be dead. And remember there won't be any until after the SSA was begun and few of the records are available online after about 2010. Early on farm families did not participate, nor did government workers. So whole groups of people will not have a record.
If the government offers the records, they would be with the National Archives and Records Administration (nara.gov).