It is possible to acclimatise to operating at high altitude without oxygen.
WW1 pilots operated regularly up to 22,000 feet in open-cockpit biplanes without oxygen, and often flew for up to 2 hours above 18,000 feet. They were very young and fit, though! Even so, it is clear from their accounts that it was physically punishing, especially in winter.
Mountaineers also are able to operate at altitudes in excess of 20,000 feet - Everest is 29,000 feet and has been scaled without oxygen.
The key is acclimatisation, in addition to cardiovascular fitness. Some people seem also to cope with / adapt better to reduced oxygen than others, even with apparently similar levels of fitness. Research is ongoing into possible genetic link to altitude capability.
FBW