[...]
It has grown 10 gigabytes in the past 6 months, do you really think it will "easily reach two hundred fifty gigabytes" in the next 6 months?
Now that we've established that the video is making predictions that are complete nonsense, lets move on to your concern that "in the near future only big companies will be mining it".
While the predictions are totally useless (simple extrapolation), it's important to note that blockchain (size) growth depends a lot on transaction activity. Transaction activity correlates with price swings (mostly upward ones). Therefore it is possible that we might see periods of (super-)exponential blockchain growth that might be problematic if a significant part of the established infrastructure isn't prepared to cope with it.
You ask "How to fix this problem in the future?"
What problem? What's wrong with only big companies mining it? That's exactly what it was intentionally designed for. From the very beginning it was specifically intended that eventually only big companies would be mining it. This isn't a "problem" that needs to be fixed. This is a "design feature" and the "intended behavior" of the system. If you didn't realize that, then the only thing that needs to be "fixed" is your understanding bitcoin.
Here is what Satoshi had to say about it himself (bold emphasis added by me):
[...]
We should not think of Satoshi as a god, who is free from any fallacy. It is generally desirable to keep the blockchain size as small as possible to keep decentralization high. Describing a large blockchain as a "design feature" or "intended behavior" is wrong, because it's not the goal of Bitcoin to max-out storage resources - a large blockchain is (currently) simply a necessity for Bitcoin to work securely.
ya.ya.yo!