What will happen to Academia in the future when it takes too long to become an Academic? Currently, here's how Academia works:


Study for many, many years, so that you have the necessary background knowledge to embark on your own research. You can still do a bit of research on the side, but mostly you are a student learning the ropes. You are (usually) not paid during this step. Even if you are, it won't be much.
Do research. You can (and probably must) still do a bit of study on the side, but mostly you are a researcher contributing to new knowledge in your field. You are paid during this step. 


This path is only tenable if the first step, the unpaid one, is relatively short, otherwise it becomes economically unfeasible to become an Academic.

But in Academia, we build upon past knowledge. Hence, it stands to reason that as time progresses, the time and effort needed to complete step 1 and acquire all the necessary background knowledge also increases. 

It increases sloooowly, but it does increase. So my question is, what happens when the time needed to complete step 1 has increased by so much that it simply isn't feasible anymore to become an Academic? Hell, what happens when the time needed to complete step 1 increases beyond the life expectancy of humans?

Again, this happens extremely slowly, but it does happen. So does this mean that in a few hundred years, Academia will be a dying field, since nobody simply has time to learn all the background knowledge?
There is one thing that you are leaving out and it is critical to understand it. Researchers work at the frontier of knowledge, the frontier is vast and growing, but an individual only works (concentrates) on a very small part of it. Getting to the "research edge" has always been hard, but no one knows everything within the known bubble of a field if the field is at all advanced. 

For example, it is no longer possible for any individual to know all of mathematics. The last time it was possible was early in the 20th century. Henri Poincaré may have been the last. There is too much there. But no mathematician worries about that. You can know a bit about a lot of things, but to work in mathematics you need to know a lot about a few things. So we specialize. 

It is also the fact that the path taken by a researcher to reach the edge (say, in mathematics) isn't the same path that was taken 50 years ago, since much of what has been learned in the past hundred years has been abstracted into the courses that fresh students take. That compression wasn't available in the past when a sub-field was fresh. 

What you describe as the necessary path isn't wrong, but you don't need to learn everything that it is possible to know in order to be effective in extending the frontier of knowledge. A researcher's path is narrow and deep. You focused on the deep, but it is the narrow that makes it feasible. 

For completeness, however, I'll also note that there are parts of academia that value the broad as much as the deep. There is room for the people who know a lot (broad) and can synthesize it for students and for the creation of the learning materials that students use to guide their own path. Such people may not be state-of-the-art researchers, but they understand the edge of the known universe well enough to describe it properly.