In some contexts, "the ones out" can refer to the people who experience financial loss in a complex financial situation.
The bet isn't clear. It hasn't been made yet. They are daydreaming about a bet, and a deal, they hope they can set up.
The point is that they hope for a deal in which they are not risking their own money. They are going to be involved in a bet against Greco, but the money is coming from "the Bostonians" and if the chess master loses, it is the Bostonians who will lose money, not the chess master.
Here is a possible way it could happen. Greco will only play if there is a bet at high stakes. The master can't afford to make that bet. He is able to find backers, the Bostonians, who are willing to make that bet so that the game can take place. The Bostonians are confident the master will win and anxious to make the bet.
The master tells them he will only play if the Bostonians take all the risk of losing, but let him keep "part of Greco's stake" if he wins.
"Hey, Bostonians, want to make some money? Greco puts up $100,000, you put up $100,000, but if I win the game and you win the bet, you have to pay me $20,000 out of Greco's stake."
Obviously this deal is favorable to the master. But someone might agree to it if they are really confident that the master can win, and don't think the master will play unless they agree to this one-sided deal.
In a bet, the "stake" is the money you are betting. Bettors want to be sure of getting paid if they win. Bettors might want to see your "stake," cash or a solid proof you really have the money. You have to get that "stake" and prove you have it before you can make the bet.
Here's another example of use. "I'm not sure this hospital bill is correct, but I'm not going to dispute it. I have health insurance, so if the hospital is overcharging, the insurance company is out, not me."