I'm out of shape and decided to start running. Made it about 400 meters yesterday before wanting to die. How do people run 5K or a marathon? That seems literally impossible from where I am right now.

Any advice for a complete beginner? Good shoes? Training plan?

The Couch to 5K plan is perfect for beginners. It alternates walking and running intervals, gradually building up. Most people can run a full 5K in 8-9 weeks following it.

Good shoes are the single most important investment. Go to a real running store and get properly fitted. Bad shoes = injuries.

Everyone feels like dying at first. Your cardiovascular system adapts faster than you'd expect. In 2-3 weeks it'll feel completely different. Trust the process.