Why is Stack Overflow's search box only big enough for two or three words? SO is a programming website, after all, isn't it? Shouldn't we have a big fat multiline text area so users can put log output or code samples in the search area? We should at least make it half as big as Google's input box.
Put the cursor in the search box and hit enter without entering any text. You get a bit bigger search box on the resulting search page with which to search using Google, Bing, or DuckDuckGo. Alternatively, you can simply go to Google and type site:stackoverflow.com (or whichever of the SE sites you care about) in the Google search box along with the rest of your query.
As far as why the box is small -- searches perform better on keywords than long strings of text. Even when I have some error output I usually distill that down to just the keywords in the message, omitting all the common words that don't help. If you feel that you regularly need a very large search box for your search text, you might find that you get better results reducing the amount of search text you use by choosing only the most unique (problem-related) words instead.
You're supposed to just use Google instead.
I'm not really a fan of this, but that's the idea here. The "powers that be" figure they can't build a better search than Google, so they don't really try to make the search all that usable.
Note that you can also use http://stackoverflow.com/search.