Perhaps no storage room is more hallowed in cinema than the 67-square-foot space known as the Criterion Closet. That tiny office closet is huge on social media thanks to criterioncollection’s “Closet Picks” videos, which launched in 2010, featuring famous actors, directors and musicians entering the space, picking movies to take home and explaining why those films mean so much to them. The videos, which now number more than 260, have become appointment viewing for movie lovers, who relish watching their favorite artists rhapsodize about cinema in unrehearsed, genuine ways. Many Criterion junkies have imagined what it would be like to hang out in that closet — to be enveloped in that cozy cocoon of great movies. For them, the Mobile Closet is the next best thing. “For the 40th anniversary, we’ve been talking about, ‘What could we do that truly engages all the people that love film?’ ” El Shami explains about the Mobile Closet’s origins. “Somebody said, almost as a joke, ‘What if we put...