Hypothetically, you could imagine that a long voyage might need a decent Netflix library, and that new media production might not be feasible en route -- perhaps this would help maintain a common language?
But by the time such a voyage is possible, content creation will likely be entirely virtual, so wouldn't need much room on the ship.
In some sense, the crew will be like a small group of survivors living in the ruins of a great empire. They see what their ancestors were capable of, but they are unable to replicate most of it. The legacy content in the old Netflix library was created by a society with the same technology but vastly superior resources. Its creators were often more talented than the best and the brightest of the current generation, and they certainly could draw on a much wider range of life experiences.
I'm willing to bet in 50 years, software will be able to fully generate new episodes of TV shows, incorporating reviews of the previous episodes. Maybe everyone watches the same generated storyline so they can share the experience and talk about it, or maybe each episode will be tailored to your specific tastes.
Point is that I doubt "running out of content" will ever be a serious concern for long-duration travel again.