Update: To my knowledge, there is no way of doing this with Flexbox. Flexbox is more concerned with horizontal layout, not vertical. I would be happy to be proven wrong, but that’s what I’ve gathered from my limited experience with it. If you want to learn more, the best article I’ve found on the matter is this: (Yes, of course it’s Chris Coyier. How’d you ever guess?)

In any case, even if you can do it with Flexbox it would be a bit of a hack, because that isn’t what Flexbox is for. There’s a much cleaner way of doing it with CSS3 columns. Here’s an example.

Browser support isn't the greatest though: Even this example doesn't seem to support Firefox, although I haven't a clue why, since FF does indeed support the respective properties, according to CanIUse.

So, a summary and TLDR: it's an admirable idea, doing this in pure CSS, but for most practical purposes is impossible at the moment. You would probably be better to go with JQuery Masonry