Just as oil and gas companies plow billions of dollars in searching for new energy reserves, big technology companies are spending lavishly on a global footprint of sophisticated computers to run every startup and corporate colossus’s business in the cloud.

Amazon.com Inc. upped the ante this week, announcing plans to plunk down a massive collection of data centers in Stockholm. It is the latest move in a high-stakes race to own the biggest piece of a market that is expected to reach into the hundreds of billions of dollars....