VENICE BEACH, Calif.—On a recent afternoon, a gymnast balanced on his hands atop a third-floor balcony railing overlooking the sidewalk.

A woman recorded the stunt on a smartphone, hoping to capture a this-is-crazy-style video that millions of people will share. That’s the kind of work done all day at 421 Media, a well-funded content firm that produces memes for consumers of Instagram, the social-media platform that in less than a decade has exceeded 1 billion monthly users.

Instagram...