Is it possible to move into air traffic control later in life? Why are there rules against training in ATC over thirty years old? How does one get started in ATC?

The reason for the ban on getting hired once you have your 31st birthday is pretty common-sense... it's because the older you are when you get hired, the less chance you have of successfully completing the training program. There were a number of studies done on this question through the 60s and 70s, and some of them are still online (http://libraryonline.erau.edu/online-full-text/faa-aviation-medicine-reports/AM71-36.pdf is an example). (If you're curious, you're looking for studies from CAMI, the FAA's medical branch that studies things like this.) When you add this lower rate of success to the declining ability to do the job that starts in the 50s, and the mandatory retirement at age 56, it becomes pretty obvious why the FAA is one of the few employers that is allowed to deliberately practice age discrimination.

The sister of a woman I worked with was an ATC. They were an Air Force family, and they were both in Air Cadets. And that is how she got into it, as part of the training they get as young people.

If you talk about Indian Air Traffic Control scene, where Airports Authority of India is the only Air Traffic Services provider in India... Then you must know that it has a strict age limit of 27-31 or 32 (depending on employment being provided in accordance to reservation categories namely General, SC/ST and OBC). People are only recruited at the starting designation. This is done to ensure organisation's hierarchical structural integrity and officers getting adequate career growth right till the time they get to their retirement age. Their is also a scientific reason to this. As per research studies in human behavior and psychology, it has been found that learning new skills keeps getting more and more difficult as one advances in age. An age old adage truly says that "There's no age to learn something new" but what it fails to tell, is that with age "learning" becomes more and more difficult. The person becomes slightly more and more presumptuous, he/she denies acceptance of their shortfalls, their mistakes. People start losing the mental flexibility that is crucial for learning a skill and when the skill is something like being able to perform Air Traffic Control  related duties with the level of perfection this job demands as a mandatory requirement... then you'll need to be in your best "learning mode" to make it happen. Age is something which will make it more and more difficult even if you want to acquire this skill with all the willingness of the world. Seeing yourself not able to learn as quickly as the next younger person besides you will only garner frustration and then even if you clear training days somehow, these days of frustration will give you a very shaky, unsure and self doubting start to a career in ATC, which in my opinion can very well end up in you feeling like living a nightmare.