You need to redefine the \emph command, and you will need the color package to provide you with a shaded box. Try this:
\documentclass{article}
\usepackage{color}
\definecolor{light-gray}{gray}{0.85}
\renewcommand\emph[1]{\colorbox{light-gray}{\texttt{#1}}}
\begin{document}
This is some \emph{marked} text.
\end{document}

However, there's a smarter way to do this that let's you use back ticks in the same way as SE does, using the newverbs and shortvrb packages:
\documentclass{article}
\usepackage{color}
\definecolor{light-gray}{gray}{0.85}
\usepackage{newverbs}
\usepackage{shortvrb}
\newverbcommand{\cverb}
{\setbox\verbbox\hbox\bgroup}
{\egroup\colorbox{light-gray}{\box\verbbox}}
\MakeSpecialShortVerb{\cverb}{\`}
%
\begin{document}
This is some `quoted` text, that allows `\TeX` macros to be shown without expansion.
\end{document}

But don't do this if you want to use the back tick for the left hand typographical quotation mark as Knuth originally intended in plain TeX.
And (as pointed out in the comments) putting things into a box like this prevents line breaks inside the box, so this approach is only suitable for short pieces of text, and you should be prepared to rewrite to avoid over- or under-full lines.
xcoloris preferred overcolorand is (at least largely) backwards compatible. – Sean Allred Sep 23 '14 at 17:21xcolorwould work just as well, butcoloris smaller, simpler, and not-yet-obsolete. Without more context or a MWE from the OP it's hard to tell which to prefer, so I went for simplicity. – Thruston Sep 23 '14 at 17:48\emphworks fine for me. The second solution does not work. I suppose the problem is that I use XeLaTeX or maybe that I use\usepackage[babel, german=quotes]{csquotes}\MakeOuterQuote{"}– Johannes Sep 23 '14 at 21:03soulpackage. – Fritz Sep 23 '14 at 22:30