Orbit can occur at any distance from Earth if you are the right speed I guess -- although below a handful of fingers you'll get significant drag from the atmosphere. Since the atmosphere doesn't just suddenly cut off at some point, it is a matter of degrees. But for example, the ISS orbits around 400km and needs to be boosted a 'few times a year.'

https://en.wikipedia.org/wiki/Low_Earth_orbit#Examples

(The Karman line is around 100km, so if we want to use 500ish km as the threshold for reasonable orbits, we really are in the ballpark of a hand's worth of fingers).

The altitude isn't really the point, so much as the trajectory -- if they miss the ground at least once I say we give them a round of applause.

Bit of a bad comparison of accomplishment. Low Earth Orbit isn't that much higher. But it is much much faster (which is what's hard). Getting into orbit is about going fast enough that the centripetal force from your velocity keeps you aloft. To get into a stable orbit at a precise target altitude requires accurate timings and controls, and lots more energy. The engineering challenge is orders of magnitude more difficult.