In your Twenty Eleven child theme (don’t edit the main theme!) create a new header.php file, copy the content from the parent theme, and replace this part:
<body <?php body_class(); ?>>
<div id="page" class="hfeed">
… with something like this:
<body <?php body_class(); ?>>
<div style="background:#345;color:#eee;padding:10px 20px;margin:0 -2em">Hello</div>
<div id="page" class="hfeed">
But move the CSS into a copy of the style.css, and adjust the styles in the media queries too, because body doesn’t have a padding in smaller windows, and you have to remove the negative margin then.