Yes, they can, but it isn't smart. If they are going to get a puppy or kitten at 70, they need to have someone lined up to take on the dog or cat FOR LIFE 5, 10, 15 years down the road. When people get that old, they rarely have the energy or life span to take on an animal that will live 10-20 years. What happens when they die? Go into a nursing home? That's when you see all sorts of older animals turn up at shelters.
When my mother died at 79, she had one seriously disturbed cat that was about 7-8 years old. It had feline hyperesthesia and could be violent. Thank goodness a relative took on that cat and kept it for 7-8 more years.
DH and I have gotten 2 kittens over the past 2 years. Since my last cat lived to be almost 20, I would not be surprised if these do too. Adding 20 years to my age, I will be pretty dang old when these cats pass, if they make it into their upper teens. We will not get any kittens when these pass, because we probably won't have another 20 years left in us and it's not fair to take on a pet you can't take care of FOR ITS LIFE.
Yes, puppies and kittens are adorable and you can't help but want them. But you must add on their expected life span to your age and see if you will be in good enough shape at that age to properly care for them. If not, don't get them. Or get an older pet that has less years left to live.