https://en.wikipedia.org/wiki/List_of_presidents_of_the_United_States_by_age
The previous record was co-set by Ford and Reagan, who were in their early 90s. Before that it was Adams (90) and Jefferson (83). Hoover also lived to 90, but I think the oldest other president at the time of Hoover's death in 1964 was Truman (80).
I've always found Hoover's long life to be particularly interesting since he really transcended eras (Van Buren is another president who seemingly lived into another era). This anecdote from Hoover's Wikipedia page is mildly amusing:
I would at this point add Carter to this as somebody who "transcended eras" -- anybody who continues living for more than 30 years post-presidency (so far, only Carter and Hoover) feels incredibly "fish out of water"-esque, and is obviously a living piece of history at a certain point. It's not too weird that Van Buren lived to see the Civil War if you know that he remained very politically active and apostatized from the Democratic Party in the 1840s (he ran an anti-slavery third-party presidential comeback campaign in 1848 that received 10% of the popular vote, but no electoral votes), so he really only lived for 14 more years after leaving politics, which doesn't seem too weird at all (though, yeah, very few prominent figures of the 1820s actually lived to see the Civil War break out as Van Buren did).
Yeah, I don't know though. I feel like to be a living piece of history you have to be somewhat reclusive and not come out much. Carter obviously doesn't fit that as you said, and given that his kids and grandkids have entered politics, Bush Sr. doesn't either.
Reagan is a good example of it, though. He was out of the limelight for years before he died for obvious reasons, and his legacy was always referred to in the past tense.