Well, as you can imagine, there are a lot of potential problems with spending time in an environment where you can't survive without life support equipment. But the good news is that scuba gear, if properly cared for, is really quite reliable, and most of the time, will give you plenty of warning before it fails. For the things that can go wrong quickly, you learn coping strategies in your open water class (and it's a good idea to practice those things from time to time afterwards, too!).
You will learn in your OW class that the depth limit for "recreational" diving is 130 feet. The stories you are reading are of dives done far deeper than that -- when you do such very deep dives, the risks are much greater. The OW depth limit is in part to make sure that, for all of us, a direct ascent to the surface (at a controlled speed) is always an option. Because of this, if you keep your wits about you and follow some very simple rules, you are very unlikely to get hurt diving. Crawl inside wrecks at 250 feet, or swim a mile back in a cave, and things are a bit different . . .
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"Sometimes, when your world is going sideways, the second best thing to everything working out right, is knowing you are loved..." ljjames