As a retired school teacher, and to become a professor at a university. Is it okay to get into a PhD program at the age of 65 and complete it to become a professor? Or is it too late and I can’t get employed at a university?
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.
It could be hard to get into a PhD program since potential advisers will be looking for students who will have many productive years ahead to establish a legacy. But you don't need a PhD to be a lecturer at most universities here in the US. You only need a master's. It's not the same as the tenure track positions, which actually have professor titles, but the students still call you professor.
I'm 66. I fell into teaching when my department chairman at University of Washington Bothell posted a request onto an IEEE email list for industry veterans to advise teams of seniors on their Capstone projects. I thought I was volunteering to do it free and was surprised it paid and even more surprised they'd pay me. With only a master's, it was never on my radar that any university would let me do this. At Washington, I was an affiliate (part-time), but that led to being recruited to a full-time three year appointment at University of Michigan. It's possible it could happen to you, too.
In most academic fields in the US it takes most people 10 years or more to go from the start of a Ph.D. program to starting a tenure-track academic position. This includes the time to get a Ph.D. plus one or two postdoctoral or visiting assistant professorship positions. If you start at 65 you're going to be looking for permanent positions when you're 75, and odds are that means you'd be looking at 10 years of preparation for a position you're likely to hold for less than five years. You would also probably need to move to a new city at least twice during those ten years. That doesn't seem like a great plan from your point of view. Also it's not clear at this point whether you will still be interested in and able to do the job in your late 70s. Lots of people at your current age are effective faculty, but the vast majority choose to retire before their late 70s.
This may be difficult for younger academicians to comprehend: There is the adage that 'Those who can, do; those who can't, teach'. Let's say that I earn a B.S. or M.A. in computer engineering and go onto to start my own company, make a gazillion dollars, sell that company and start another. I sell that company and now have a net worth of twenty billion dollars. I'm sixty years old, I have the time and money. I've made my fortune, set up a foundation to do good works, and am now eager to explore other areas of life. I think, why not get that PhD? It has nothing to do with teaching; it's about challenge, accomplishment, and completion. And who knows, maybe some university will hire me--not because I'm young, but because I have wisdom and life experience.