> All this was stored on paper
Rather, as also mentioned in the answers, on Hollerith cards, 5-digit encoded (2 for the year, 3 for the day of year). Punch-card appliances were very much a thing. (WWII logistics were run by punch-cards.)
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They used IBM cards, not Hollerith cards (they used IBM 077 collators and 405 accounting machines).
They were coded with 6 digits: 2 for month, 2 day and 2 year according to the SSA.
Notably, this is from December 1955, already on computers. There's no mention of how data was handled before this. (Also, pre-WWII, people used do talk of "Hollerith cards" in a general way – and the system started in the 1930s.)
It's mentioned on IBM's site:
https://www.ibm.com/history/social-security-act
https://www.ibm.com/history/punched-card#The+Social+Security...
And the SSA site mentions using IBM machines for it in 1936. IBM machines only used IBM cards:
https://www.ssa.gov/policy/docs/ssb/v69n2/v69n2p55.html
The checks used to pay people were also IBM punchcards.
Hollerith founded the Tabulating Machine Company, which became the Computing-Tabulating-Recording Company (CTR), which became International Business Machines Corporation (IBM). These are the same thing, admittedly adding other businesses along the way. (Hence, IBM includes the invention of the tabulating punch-card in its company history.) So, after 1924, it's IBM, but this didn't stop people from talking about Hollerith cards, even in the 1950s. Here, it was used to designate the pre-computer era use with punch-card appliances.
I didn't realize IBM called their own cards Hollerith cards.
They changed the format to 80x10 with rectangular punches in 1928, naming it the IBM Computer Card. They modified it to 80x12 in 1930. IBM themselves distinguish between Hollerith's cards and IBM cards, but I suppose they were the same physical size and old names die hard.