Html – ny way to have a fieldset width only be as wide as the controls in them

2015 update: the flexbox approach

There are two other answers briefly mentioning flexbox; however, that was more than two years ago, and they don't provide any examples. The specification for flexbox has definitely settled now.

Note: Though CSS Flexible Boxes Layout specification is at the Candidate Recommendation stage, not all browsers have implemented it. WebKit implementation must be prefixed with -webkit-; Internet Explorer implements an old version of the spec, prefixed with -ms-; Opera 12.10 implements the latest version of the spec, unprefixed. See the compatibility table on each property for an up-to-date compatibility status.

(taken from https://developer.mozilla.org/en-US/docs/Web/Guide/CSS/Flexible_boxes)

All major browsers and IE11+ support Flexbox. For IE 10 or older, you can use the FlexieJS shim.

To check current support you can also see here: http://caniuse.com/#feat=flexbox

Working example

With flexbox you can easily switch between any of your rows or columns either having fixed dimensions, content-sized dimensions or remaining-space dimensions. In my example I have set the header to snap to its content (as per the OPs question), I've added a footer to show how to add a fixed-height region and then set the content area to fill up the remaining space.

html,
body {
  height: 100%;
  margin: 0;
}

.box {
  display: flex;
  flex-flow: column;
  height: 100%;
}

.box .row {
  border: 1px dotted grey;
}

.box .row.header {
  flex: 0 1 auto;
  /* The above is shorthand for:
  flex-grow: 0,
  flex-shrink: 1,
  flex-basis: auto
  */
}

.box .row.content {
  flex: 1 1 auto;
}

.box .row.footer {
  flex: 0 1 40px;
}

<!-- Obviously, you could use HTML5 tags like `header`, `footer` and `section` -->

<div class="box">
  <div class="row header">
    <p><b>header</b>
      <br />
      <br />(sized to content)</p>
  </div>
  <div class="row content">
    <p>
      <b>content</b>
      (fills remaining space)
    </p>
  </div>
  <div class="row footer">
    <p><b>footer</b> (fixed height)</p>
  </div>
</div>

In the CSS above, the flex property shorthands the flex-grow, flex-shrink, and flex-basis properties to establish the flexibility of the flex items. Mozilla has a good introduction to the flexible boxes model.

Just create a wrapper <div> with a percentage value for padding-bottom, like this:

.demoWrapper {
  padding: 10px;
  background: white;
  box-sizing: border-box;
  resize: horizontal;
  border: 1px dashed;
  overflow: auto;
  max-width: 100%;
  height: calc(100vh - 16px);
}

div {
  width: 100%;
  padding-bottom: 75%;
  background: gold; /** <-- For the demo **/
}

<div class="demoWrapper">
  <div></div>
</div>

It will result in a <div> with height equal to 75% of the width of its container (a 4:3 aspect ratio).

This relies on the fact that for padding :

The percentage is calculated with respect to the width of the generated box's containing block [...] (source: w3.org, emphasis mine)

Padding-bottom values for other aspect ratios and 100% width :

aspect ratio  | padding-bottom value
--------------|----------------------
    16:9      |       56.25%
    4:3       |       75%
    3:2       |       66.66%
    8:5       |       62.5%

Placing content in the div :

In order to keep the aspect ratio of the div and prevent its content from stretching it, you need to add an absolutely positioned child and stretch it to the edges of the wrapper with:

div.stretchy-wrapper {
  position: relative;
}

div.stretchy-wrapper > div {
  position: absolute;
  top: 0; bottom: 0; left: 0; right: 0;
}

Here's a demo and another more in depth demo