The interpretation of the assignment that seems to make most sense is that you are required to use XHTML linearization of HTML5, also known as XHTML5. This simply means that you use HTML5 like anyone else but do that using general XML principles.
In the example case, this would mean the following markup:
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<title></title>
</head>
<body id="index" class="home">
<header id="banner" class="body">
<h1><a href="#">Header1 </a></h1>
<nav><ul>
<li class="active"><a href="#">home</a></li>
<li><a href="#">portfolio</a></li>
<li><a href="#">blog</a></li>
<li><a href="#">contact</a></li>
</ul></nav>
</header>
</body>
</html>
The XHTML 1.0 doctypes are something quite different. They define fixed versions of HTML, so you cannot use, in the static markup, anything not allowed by those versions, i.e. anything that is new in HTML5 as compared with XHTML 1.0 and HTML 4.01 (which is what “HTML5 feature” probably means in the assignment). The requirement “XHTML compliant (Strict or Transitional)” is obscure, but if it is meant to refer to XHTML 1.0 specifically, then the assignment is self-contradictory (unless you are supposed to use client-side scripting to get to “HTML5 features”).
(This answer was largely rewritten thanks to @Alohci’s comments.)
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