There is no age limit. But, I would add that it all depends on the new career you have chosen; the current skill set you have that is related to the new career; your dedication to making the transition; and your age. For example, if you are a 60 year old Factory worker with no medical background, you might think twice of transitioning to be a doctor. However, if you are a factory worker who was a medic while in the Army, transitioning to be an EMT might fit quite well. When considering a career change, first take an inventory of your current skills, then look to see what other careers might require those skills. Once you have selected a new career goal investigate what the requirements are for it. Then map out a plan on how you will obtain those skills and the dates by which you plan to achieve them. For example if your current background is in Customer Service, and you want to transition into the Information Technology field, you might consider seeking a Help Desk position. To do that you might start by taking classes at a local college; perhaps some specialized training to get certified in PC repair.
Whatever new career you choose, recognize the amount of planning, work, and dedication required by you to accomplish the objectives that will lead to your new career.