A: What are your memories of the U. S. - Soviet Union game in 1980, Seth? ;-)
B: I was all of six months old that day. ;)
Does he mean he's exactly six months old that day?
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When used with age, the all of means that someone is or was too young for something or doing something.
all of - As much as (often used ironically of an amount or quantity considered small by the speaker)
This means, he was too young to have memories.