This sounds like it came from a movie.
"Well, after (or on) a night like this, the minute I sat down I ordered a martini, straight up and dry as a bone."
I'm guessing that he's had a difficult night and needs a drink to relax. Straight up is a way of ordering a drink, where the alcohol is iced, and then the ice is strained out, so the alcohol is served without ice but chilled. I don't actually know what "dry" means. I think it means that there is dry vermouth in the drink instead of sweet vermouth.
"Martinis are in, aren't they?"
This can be translated as "Are martinis popular right now?"
"Far as I'm concerned, they never went out."
This can be translated as "As far as I'm concerned, martinis have always been popular."