Using the following styles:
.divideMe{
width:100px;
word-wrap: break-word;
}
<div class='divideMe'>reallyreallyreallyreallyreallyreallylongstring</div>
It would output
reallyreallyreally
reallyreallyreallyl
ongstring
Though it doesn't really break the word in chunks so you can still select it by double (triple) clicking it. I'm not sure of how browser compatible this is but it seems to work fine on the browsers I have available. Keep in mind that it's a CSS3 property.
Tested on IE 6,7,8 works fine. Also fine on FF 3.6
Another solution would be to hide the overflow and show a scroll bar if you want to keep it in one line. But since your first approach was to show the whole string instead of hiding it, I think this may be a good approach.