The memo protocol stores data directly in the blockchain. Storing text shouldn't be an issue even in the long run, but video etc. must be stored externally and linked.
It's important to note that Bitcoin users do not need to store the entire blockchain, and financially-irrelevant data can be pruned. So there's flexibility when it comes to storage.
Using the Bitcoin blockchain as a communication medium does come with storage costs, but those who want to use Bitcoin solely as money aren't really forced to bear those costs.