The Slot House in Dobuita (between Knocking Motorcycles and Bar Yokosuka) is the only place in Shenmue where you can make money, instead of just spending it - although since it's fairly addictive once you get started, you can walk away considerably lighter as well.
Winning is easier if you've gone to Lapis the fortune teller to get your lucky number, but it's not impossible if you haven't.
Playing The Slots
Different machines in the Slot House require different minimum bets; wagers go a from ¥1 to a ¥20 minimum. The advantage of higher-bet machines is that you'll win more when you're lucky - the advantage of lower-bet machines is that you won't lose as much when you're not.
You have to select how much to convert to tokens (in ¥100 increments), then you'll get to place your bet.
Betting
You can place up to 5 bets on each machine - one for each winning "line". The more bets you place the more chance you have to win, but the more you stand to lose.
Move Ryo's hand over the "bet" button and press 
to add a bet to the machine.
Spin!
There's two ways to spin the machine - either move Ryo's hand over the "Play" button, or press right to grab the handle. In both cases, pressing will start the wheels spinning, leaving your money to Lady Luck.
Cashing Out
When you're done, select and press the "Pay Winnings" button, and you'll get your winnings - or at least, what you haven't lost. If you plan to come back, you can leave your tokens with the cashier for later.