Responsive video iframes (keeping aspect ratio) with only CSS?

Here is a Fiddle for a solution, that is based on a CSS2 secret: https://jsfiddle.net/59f9uc5e/2/

<div class="aspect-ratio">
  <iframe src="" width="550" height="275" frameborder="0"></iframe>
</div>

<style>
/* This element defines the size the iframe will take.
   In this example we want to have a ratio of 25:14 */
.aspect-ratio {
  position: relative;
  width: 100%;
  height: 0;
  padding-bottom: 56.25%; /* The height of the item will now be 56.25% of the width. */
}

/* Adjust the iframe so it's rendered in the outer-width and outer-height of it's parent */
.aspect-ratio iframe {
  position: absolute;
  width: 100%;
  height: 100%;
  left: 0;
  top: 0;
}
</style>

It is explained by how percentage-values for padding are handled:

The percentage is calculated with respect to the width of the generated box’s containing block, even for ‘padding-top’ and ‘padding-bottom’.

https://www.w3.org/TR/CSS2/box.html#padding-properties