First and foremost requirement is that you mustn't be prone to travel sickness. Lunar modules are notoriously cramped and you don't want to start a five-year mission with your suit filled to eye-level with last night's celebratory curry.
To become an astronaut you have to forfeit the right to an interesting haircut. Whatever style you choose, it's going to end up the same shape as the inside of a helmet, so it's best to opt for a haircut that looks like the inside of a helmet before you start training. Some people think that to be an astronaut you have to be obsessed with space and planets since the age of four. Wrong. You will grow up to be a nerd, not an astronaut. Instead, you grow up fairly normally, be a test pilot and then study something epicly dull such as material sciences or geology. This is important because trips to the planets are a lot like trips to the seaside in that you bring back a lot of rock.
Astronauts aren't just scientists; they also need to have an ear for a good phrase. When you put your foot on the moon for the first time, it's no good if the best you can do is, "Hey, look what I've just stepped on." However, when you get back, you need industrial-strength modesty to say, "Heck, we was just doing our job."
One of the most important characteristics necessary to be an astronaut is a complete imagination bypass. This means when you're in space you don't spend all your time peering out of the window at the beautiful, fragile earth saying "Wow, that is awesome!" and instead you concentrate on your experiment on the effects of zero gravity on cress.
You can do a certain amount of self-selection. For example, you're not likely to be selected if you don't like confined spaces and have been convicted of an air rage incident. Also, you won't get far if you can't control your Metro above 40mph and have trouble getting the nozzle in the tank (space docking happens at about 60,000 mph). Members of the Flat Earth Society need not apply.
Finally, if you meet travellers from another planet, don't fear that they are bent on galactic domination. They too were selected because they studied rocks at college and have haircuts that fit inside helmets.