This design is a canister based on the cylinder-shaped parking spot in the Batcave seen in Batman: The Animated Series. A cover slides out of the way, allowing access to trays that swing out for accessing the discs. There is probably enough room for the complete series and all of the related films.
The trays swinging out was inspired by Batman throwing a Batarang. I suppose the Batmobile could rotate on a revolving platform (would need batteries), much like in the show.