How do I use Google Takeout to download a copy of all my Google data?

I want to back up my Gmail, Google Photos, and Drive files before cleaning up my account. I've heard Google Takeout can export everything.

Step-by-step directions for finding Google Takeout, choosing which services to export, and actually getting the download.

I found takeout.google.com but the page lists dozens of products with checkboxes, different export formats, and delivery options. I selected everything and clicked export, but nothing downloaded — I'm not sure where the files went or how long it takes.