Most people consider space to simply be the region above the Earth's atmosphere. But, there is no tangible boundary because the atmosphere gradually thins with increasing altitude. Traces of the gasses we breathe can be found over 100 miles above the earth.
Many different definitions exist.
The United States awards astronaut status to anyone who flies above 50 miles in altitude. Whichever definition you prefer, it is far, far above the level that humans can breathe. Updated July 5, 1995.