To enroll in to a PhD Program at the age of 65
The realistic answer is: you can enroll in a PhD program at age 65. However, you are highly unlikely to get a tenure-track professorship after finishing a PhD program at that age. The big reason for this is that you're already above the retirement age, and many schools do have rules about how old you can be and still advise students. Moreover, you probably wouldn't represent a good "long-term investment." 

On the other hand, you may be able to obtain a position as a part-time lecturer or adjunct faculty—in other words, for positions where you are not directly supervising PhD candidates.