Can I set the height of a div based on a percentage-based width?

This can be done with a CSS hack (see the other answers), but it can also be done very easily with JavaScript.

Set the div's width to (for example) 50%, use JavaScript to check its width, and then set the height accordingly. Here's a code example using jQuery:

$(function() {
    var div = $('#dynamicheight');
    var width = div.width();
    
    div.css('height', width);
});

#dynamicheight
{
    width: 50%;
    
    /* Just for looks: */
    background-color: cornflowerblue;
    margin: 25px;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>

<div id="dynamicheight"></div>

If you want the box to scale with the browser window on resize, move the code to a function and call it on the window resize event. Here's a demonstration of that too (view example full screen and resize browser window):

$(window).ready(updateHeight);
$(window).resize(updateHeight);

function updateHeight()
{
    var div = $('#dynamicheight');
    var width = div.width();
    
    div.css('height', width);
}

#dynamicheight
{
    width: 50%;
    
    /* Just for looks: */
    background-color: cornflowerblue;
    margin: 25px;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>

<div id="dynamicheight"></div>

<div><p>some unnecessary content</p></div>

div{
    border: 1px solid red;
    width: 40%;
    padding: 40%;
    box-sizing: border-box;
    position: relative;
}
p{
    position: absolute;
    top: 0;
    left: 0;
}

For this to work i think you need to define the padding to ex. top? like this:

<div><p>some unnecessary content</p></div>

div{
    border: 1px solid red;
    width: 40%;
    padding-top: 40%;
    box-sizing: border-box;
    position: relative;
}
p{
    position: absolute;
    top: 0;
    left: 0;
}

anyways, thats how i got it to work, since with just padding all arround it would not be a square.

This can actually be done with only CSS, but the content inside the div must be absolutely positioned. The key is to use padding as a percentage and the box-sizing: border-box CSS attribute:

div {
  border: 1px solid red;
  width: 40%;
  padding: 40%;
  box-sizing: border-box;
  position: relative;
}
p {
  position: absolute;
  top: 0;
  left: 0;
}

<div>
  <p>Some unnecessary content.</p>
</div>

Adjust percentages to your liking. Here is a JSFiddle