What is the real-world significance of the Bekenstein bound?

How could any information be stored in a hydrogen atom?

Your sources aren't talking about storing information in a hydrogen atom. They're talking about storing information in an amount of space whose volume is the same as the volume of a hydrogen atom.

What is the real-world significance of the Bekenstein bound?

If "real world" means practical, then the answer is that the Bekenstein bound has no real-world significance. The WP article is being silly by applying it to computer science. They aren't referring to computer science in the sense of actual computer hardware. They're just saying that computer science deals with the storage and manipulation of information, and this is an ultimate bound on that.

If you take some matter and compress it so much that it forms a black hole, you've hidden away the information contained in that matter. It's behind an event horizon and can't be retrieved. A black hole exactly saturates the Bekenstein bound. If you want to take the same amount of information and compress it without making it inaccessible behind an event horizon, you're going to have to compress less than the limit specified by the Bekenstein bound.