How to prevent long words from breaking my div?

Two fixes:

1. overflow:scroll -- this makes sure your content can be seen at the cost of design (scrollbars are ugly)
2. overflow:hidden -- just cuts off any overflow. It means people can't read the content though.

If (in the SO example) you want to stop it overlapping the padding, you'd probably have to create another div, inside the padding, to hold your content.

Edit: As the other answers state, there are a variety of methods for truncating the words, be that working out the render width on the client side (never attempt to do this server-side as it will never work reliably, cross platform) through JS and inserting break-characters, or using non-standard and/or wildly incompatible CSS tags (word-wrap: break-word doesn't appear to work in Firefox).

You will always need an overflow descriptor though. If your div contains another too-wide block-type piece of content (image, table, etc), you'll need overflow to make it not destroy the layout/design.

So by all means use another method (or a combination of them) but remember to add overflow too so you cover all browsers.

Edit 2 (to address your comment below):

Start off using the CSS overflow property isn't perfect but it stops designs breaking. Apply overflow:hidden first. Remember that overflow might not break on padding so either nest divs or use a border (whatever works best for you).

This will hide overflowing content and therefore you might lose meaning. You could use a scrollbar (using overflow:auto or overflow:scroll instead of overflow:hidden) but depending on the dimensions of the div, and your design, this might not look or work great.

To fix it, we can use JS to pull

This is a complex issue, as we know, and not supported in any common way between browsers. Most browsers don't support this feature natively at all.

In some work done with HTML emails, where user content was being used, but script is not available (and even CSS is not supported very well) the following bit of CSS in a wrapper around your unspaced content block should at least help somewhat:

.word-break {
  /* The following styles prevent unbroken strings from breaking the layout */
  width: 300px; /* set to whatever width you need */
  overflow: auto;
  white-space: -moz-pre-wrap; /* Mozilla */
  white-space: -hp-pre-wrap; /* HP printers */
  white-space: -o-pre-wrap; /* Opera 7 */
  white-space: -pre-wrap; /* Opera 4-6 */
  white-space: pre-wrap; /* CSS 2.1 */
  white-space: pre-line; /* CSS 3 (and 2.1 as well, actually) */
  word-wrap: break-word; /* IE */
  -moz-binding: url('xbl.xml#wordwrap'); /* Firefox (using XBL) */
}

In the case of Mozilla-based browsers, the XBL file mentioned above contains:

<?xml version="1.0" encoding="utf-8"?>
<bindings xmlns="http://www.mozilla.org/xbl" 
          xmlns:html="http://www.w3.org/1999/xhtml">
  <!--
  More information on XBL:
  http://developer.mozilla.org/en/docs/XBL:XBL_1.0_Reference

  Example of implementing the CSS 'word-wrap' feature:
  http://blog.stchur.com/2007/02/22/emulating-css-word-wrap-for-mozillafirefox/
  -->
  <binding id="wordwrap" applyauthorstyles="false">
    <implementation>
      <constructor>
        //<![CDATA[
        var elem = this;

        doWrap();
        elem.addEventListener('overflow', doWrap, false);

        function doWrap() {
          var walker = document.createTreeWalker(elem, NodeFilter.SHOW_TEXT, null, false);
          while (walker.nextNode()) {
            var node = walker.currentNode;
            node.nodeValue = node.nodeValue.split('').join(String.fromCharCode('8203'));