In some societies, the ability to handle hard liquor - to drink shots of very strong alcohol without flinching and without appearing affected - is taken as a marker of masculinity, of being a man.
In this scene, Quint supplies his own home brew- - by the accounts in the scene, fiercely strong and almost undrinkable - to, first, his estranged daughter, who narrates the scene, and then to his other guests. His daughter of course downs the drink and doesn't flinch: she grew up with it, probably. But the men don't ; they sip at the drink rather than down it in a gulp, and try to hide their attempts to get rid of their shots while Quint isn't looking. Of course Quint sees: he is using the alcohol to get a measure of, to inform an opinion about the qualities of the people who have come to see him. He is not impressed.
"Can't get a good man under 60" -- in this town, you can't find a good man who is younger than 60 years old (that will include his visitors, but not him.) "All gone at least 35 years." I read this as US, informal for "they have all been gone since at least 35 years ago." I suspect the back story in the novel, and probably the film - I think the implication is that this small (fictional) seaside town, a failing resort town dependent on summer tourism from New York (and not yet home to a mega-casino, organised crime and the visitors that those might bring), has been losing its best men to the city or to other migration for years. There's nothing in this small, failing resort to keep a good man - a tough, ambitious, enterprising, reliable guy, the kind of guy who could down a shot of firewater in one and not blink, who could stand beside you in a firefight or a battle with a giant killer shark and who you could trust with your life - those kinds of guys have all been gone for years. No wonder his daughter left town ...
https://taskandpurpose.com/entertainment/the-darkest-moment-in-jaws-had-nothing-to-do-with-that-shark