Verbatim in tikz node

One way is to save one or more verbboxes prior to entering the tikz. Then, recall them as needed.

\documentclass{article}

\usepackage{tikz,verbatimbox}

\usetikzlibrary{positioning}

\begin{document}
\begin{myverbbox}{\sTWO}
This i$ my \verbatim
  environment
\\##@
\end{myverbbox}

\begin{tikzpicture}
\node (s1) {$a$};
\node (s2) [right=of s1] {$b$};
\node (s3) [above right=0cm and 1cm of s2] {$c$};
\node (s4) [below right=0cm and 1cm of s2] {$d$};

\node[right=6cm of s2] {\sTWO};

\path (s1) edge (s2);
\path (s2) edge (s3);
\path (s2) edge (s4);
\end{tikzpicture}

\end{document}

Here is a version that demonstrates multiple verbboxes, the second one in color and \footnotesize:

\documentclass{article}

\usepackage{tikz,verbatimbox}

\usetikzlibrary{positioning}

\begin{document}
\begin{myverbbox}{\sTWO}
This i$ my \verbatim
  environment
\\##@
\end{myverbbox}
{\color{red}
\begin{myverbbox}[\footnotesize]{\sTWELVE}
Another
 verbatim &$#^_ environment
\end{myverbbox}
}

\begin{tikzpicture}
\node (s1) {$a$};
\node (s2) [right=of s1] {$b$};
\node (s3) [above right=0cm and 1cm of s2] {$c$};
\node (s4) [below right=0cm and 1cm of s2] {$d$};

\node[right=6cm of s2] {\sTWO};

\path (s1) edge (s2);
\path (s2) edge (s3);
\path (s2) edge (s4);

\node (s11) [below=2cm of s1] {$e$};
\node (s12) [right=of s11] {$f$};
\node (s13) [above right=0cm and 1cm of s12] {$g$};
\node (s14) [below right=0cm and 1cm of s12] {$h$};

\node[right=6cm of s12] {\sTWELVE};

\path (s11) edge (s12);
\path (s12) edge (s13);
\path (s12) edge (s14);
\end{tikzpicture}

\end{document}

You don't really need the verbatim text inside a node at all. If you take it out of the tikzpicture, you can just put it into a minipage or something to align it.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
  \node (s1) {$a$};
  \node (s2) [right=of s1] {$b$};
  \node (s3) [above right=0cm and 1cm of s2] {$c$};
  \node (s4) [below right=0cm and 1cm of s2] {$d$};
  \draw (s1) -- (s2) edge (s3) -- (s4);
  \path [->] (s2 -| s4.east) edge ++(10mm,0);
\end{tikzpicture}
\begin{minipage}[c]{.5\linewidth}
\begin{verbatim}
  Some verbatim stuff
  in a
  minipage
\end{verbatim}
\end{minipage}
\end{document}

While TikZ nodes support \verb in the node text they don't support the verbatim environment (for reasons I am not completely clear on).

But just for fun, how about rolling your own verbatim environment?

Here's a minimal-ish implementation which can automatically determine the line width (using the auto option). Alternatively the text width option to the node can be given which will wrap long lines. Otherwise you get one long line.

\documentclass[tikz,border=5]{standalone} 
\usetikzlibrary{positioning}
\makeatletter
\def\makeother#1{\catcode`#1=12\relax}
\def\makeactive#1{\catcode`#1=13\relax}
{
\catcode`\|=\catcode`\\
\catcode`\[=\catcode`\{
\catcode`\]=\catcode`\}
|makeother\\
|makeother|{
|makeother|}
|makeactive|^^M%
|long|gdef|docollect#1^^M\end{vrbtm}[|aftercollect[#1]]%
]
{
\makeactive\ %
\gdef\dospace{\makeactive\ \def {\spaceaction}}
}
{
\makeactive\^^M%
\gdef\doreturn{\makeactive\^^M\def^^M{\returnaction}}%
\gdef\beforecollect{\@ifnextchar^^M{\afterassignment\docollect\let\token=}%
  {\docollect}}%
}

\def\spaceaction{\hskip1ex}
\def\returnaction{\par\leavevmode}

\newbox\vrbbxa
\newbox\vrbbxb

\def\autobeforetypeset{%
  \let\returnaction=\autoline%
  \setbox\vrbbxa=\hbox\bgroup\ignorespaces}

\def\autoline{%
    \unskip%
  \egroup%
  \autoaddtobox%
  \setbox\vrbbxa=\hbox\bgroup\ignorespaces}

\def\autoaddtobox{%
  \ifdim\wd\vrbbxb<\wd\vrbbxa%
    \let\tmpbx=\vrbbxa%
  \else%
    \let\tmpbx=\vrbbxb%
  \fi%
  \setbox\vrbbxb=\vbox{\hsize=\wd\tmpbx%
    \ifvoid\vrbbxb\else\box\vrbbxb\fi\box\vrbbxa}%
}

\def\autoaftertypeset{%
   \unskip%
  \egroup%
  \autoaddtobox%
  \box\vrbbxb%
}

\def\beforetypeset{\noindent}
\long\def\typeset#1{#1}
\def\aftertypeset{\unskip}

\pgfkeys{vrbtm/.cd,
  auto/.code={%
    \let\beforetypeset=\autobeforetypeset
    \let\aftertypeset=\autoaftertypeset
  }
}
\def\vrbtm{\@ifnextchar[\vrbtmopts{\vrbtmopts[]}}
\def\vrbtmopts[#1]{%
  \begingroup%
    \pgfkeys{vrbtm/.cd,#1}%
    \let\do=\makeother%
    \dospecials%
    \dospace%
    \doreturn%
    \beforecollect}

\def\aftercollect#1{%
  \beforetypeset%
  \typeset{#1}%
  \aftertypeset%
  \endgroup%
  \end{vrbtm}%
}
\begin{document}

\begin{tikzpicture}[every node/.style={font=\ttfamily}]

\node[draw] (A) {%
\begin{vrbtm}[auto]
Here i$  $ome 
 verb&tim_text

with 
{automat~c} \line lengt# c%lcul%tion
\end{vrbtm}
};

\node[draw, below=of A, text width=4cm] (B) {%
\begin{vrbtm}
Here i$  $ome 
 verb&tim_text

with 
{te#t width} and word wrap
\end{vrbtm}
};

\node[draw, below=of B] (C) {%
\begin{vrbtm}
Here i$ $ome 
 verb&tim_text

 without
 {anything} \cle^er
\end{vrbtm}
};
\end{tikzpicture}
\end{document}