How old does someone have to be in order for it to be considered as them dying of old age?

There is no exact cutoff for this, but typically people will only say that a person "died from old age" if they were over 60 years old at the time of death.

There is not a specific age. Dying of old age is just a saying for when an older person's body becomes too weak to continue functioning. They don't die of illness or injury. They just fade out.