I’ve read that “humans have survived for two years at 5,950 m (19,520 ft, 475 millibars of atmospheric pressure), which is the highest recorded permanently tolerable altitude” and that “Daulat Beg Oldie...has an airstrip at an altitude of 5065 meters (16,614 feet), which is the world's highest airstrip.” I can’t seem to find any record of the highest horizontal take-off in history. I really want to know if there is an altitude at which humans can survive but take-off is impossible. Or if there are any other conditions at which humans could survive naturally but where take-off or flight is impossible for some reason other than temperature, weather or terrain.
Like most hypothetical questions of this nature, the answer is "it depends". Given enough runway at a light loading just about any aircraft with a service ceiling above 20K ought to theoretically be able to take off. One of the issues you might run into would be exceeding max tire speed.
The question of if it is possible is moot. I once asked a racecar builder what was the maximum amount of horsepower you could put in his car. His answer was, “How much money do you have?”. In other words, anything is possible with the right amount of time, resources, and desire.
In the case of your question, it is not about whether an airplane can do it. It is not even about whether a current military, cargo, passenger, jet, prop-driven, or currently produced plane can do it. It is more about where can you do it. You would need the proper facilities (long enough runway). A helicopter has already exceeded your altitude in question due to not having the runway restriction.