Well, you could put the content in a div, and then limit the height of the div to force the browser to place a scrollbar there (to scroll within the div). This way, the page fits within the browser window but the div containing the content has its very own scroll bar.
For example, here is the style from Stack Overflow's very own Related Questions list when you ask a question:
"height:150px; overflow-y:scroll; overflow-x:clip;"
As for styling, I'm not sure that's possible with this workaround.
So basically, you would have this in your page
––––––––––––––––––––
| Header |
|------------------|
| |
| Div |
| with limited |
| height |
| |
––––––––––––––––––––
Note that this method will make your header appear constantly (unless the header+div is greater than window size, which would cause two scrollbars), since the header's not actually being scrolled past.