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.