According to Universe Today, space begins where the Earth's atmosphere stops. However, the Earth's atmosphere does not stop abruptly. The Earth's atmosphere gradually grows thinner with increasing distance from the surface of the planet.
At an altitude of about 50 to 80 miles, conventional airplanes cannot fly without traveling at orbital velocity. Even at an altitude of about 250 miles above the surface of the Earth, which is the altitude at which the International Space Station orbits the Earth, there are enough air molecules to slow the craft down over time. Periodically, the space station must fire its thrusters to avoid spiraling into the Earth.