Average (arithmetic mean) life expectancy was a lot lower because it was heavily skewed by infant mortality. Up until very recently, many more babies died within their first few years of life. We've dramatically improved our ability to produce and distribute food and medicine, which has had a profound effect on infant survivability. But we haven't actually lengthened the average life expectancy of an adult who survives the first few years of childhood by all that much.

Point is, plenty of people have been living into their 70s, 80s, and 90s for as long as we've existed as a species. That said, the prevalance of diseases like Alzheimer's might be a fairly modern aberration, perhaps owing to lifestyle, environmental, and dietary factors.