(theoretically when a person is too old to work. This used to be 60 for the U.K. state pension. Years ago they were called ‘old age pensioners’. That age is increasing towards 68. But many retire earlier; pilots and people in the armed forces, for example. Roman soldiers in the empire served for 25 years. So retirement would have been around 50 or so, and the Latin word for ‘old’ (senex) was for those too old to be soldiers.
The age at which people feel or seem old varies. But I suggest that it begins when we (yes, I am old) when we start to realise there are more and more things we cannot do that we could do before. These may be physical. I used to be able to amuse my kids by fully exhaling and staying on the bottom of the pool and staying there. Now I just float back up: my body is less dense than it was. Then there was when I couldn’t stand on my head any more or do a ‘spider’ though I was good at gymnastics at school.. and though I can still run down stairs and climb ladders and trees, I am much more cautious now, because I am aware that a fall is more likely to do serious damage that would be slower to heel. Age may show up in other ways too: a less efficient short term memory, for example. But I should propose 60 as a sensible age after which old people could be said to be old: perhaps 65.; certainly 70.