Pay the Russians
As yet, the only people to go to space who didn’t go through years of training with a government agency are the handful of tourists to launch to the International Space Station in a Soyuz spacecraft. Multimillionaire space tourist Dennis Tito pioneered this method, paying a reported $20 million back in 2001 for the pleasure of spending eight days on the ISS.
Since then, six more wealthy individuals have spent between $20 million and $40 million to get to space. Singer Sarah Brightman is reportedly in talks to go to the ISS next year for $50 million.
Odds of becoming an astronaut this way: 1 in 72,000 (.0014 percent)*
According to Merrill Lynch, there are about 95,000 people worth $30 million or more[pdf] in the world (as of 2009), and technically, any one of them could decide to liquidate all or most of their assets to get to space (though they still have to survive 8 months of training at Star City in Russia). So your odds of getting to space this way are roughly the same as your odds of having $30 million. With 6.8 billion people in the world[pdf] (in 2009), that puts your chances of even being considered at around 1 in 72,000 (.0014 percent).
Dennis Tito on left. Image: NASA