For half a century, Gursaran Pran Talwar has been developing what he hopes will be the next big thing in birth control. A nonagenarian who was once the director of India’s National Institute of Immunology, Talwar envisions bringing to market a new form of contraception that could block pregnancy without the usual trade-offs—an intervention that’s long-acting but reversible; cheap, discreet, and easy to administer; less invasive than an intrauterine device and more convenient than a daily pill. It would skip messy, sometimes dangerous side effects, such as weight gain, mood swings, and rare but risky blood clots and strokes. It...