FancyBox cannot strip out the header, sidebars and footer. I see only two possible alternatives:
1. Include the form inside a hidden div on the same page and call it as inline content (instead of another page as iframe content). This basically means you wrap the form inside <div style="display:none;" class="fancybox-hidden"><div id="myform" width="xxx" height="xxx"> ... </div></div> and make the link similar to <a href="#myform" class="fancybox">open my form</a>
2. Create a complete new custom template file (that does not call the regular header.php, sidebar.php and footer.php) inside your theme folder and make the contact form page use that template... Start reading more on http://codex.wordpress.org/Theme_Development#Custom_Page_Templates
If you're new to themeing, my advise would be option 1. And if you use a form plugin like Contact Form 7, it's really not difficult recreating that same form as you are already using on the regular contact page :)