Source: cracked.com

As part of the New Deal the first Social Security numbers were not issued until November 1936. https://www.ssa.gov/history/ssn/firstcard.html

The date of birth of Steve Rogers, aka Captain America, is reported as various dates in the 1920s. At his age, even if he has never paid Medicare payroll taxes, he would be eligible for both Medicare Part A and Part B coverage. https://www.medicare.gov/eligibilitypremiumcalc/

As a veteran, Captain America may also be eligible for other benefits.