One of the things you hear a lot when you talk to entrepreneurs and venture capitalists is that they want to change the world. Some of them even mean it. One of those people is Bryan Johnson, who has committed $100 million—a substantial portion of his personal fortune—to a new venture-capital fund aimed at what Mr. Johnson himself calls “crazy” projects.

The goals of the startups Mr. Johnson is backing are varied. They include an effort to capture an asteroid and mine it, a company that wants to meaningfully increase both...