TikZ block diagram with multiple arrows not from centre of block

Two options; using anchors and some shifts, and using the <name>.<angle > syntax:

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\node[draw,minimum size=2cm] (x) {X};
\draw[->] ([yshift=-10pt]x.west) -- node[fill=white] {a} +(-1cm,0pt);
\draw[->] ([yshift=10pt]x.west) -- node[fill=white] {b} +(-1cm,0pt);
\draw[->] (x.120) -- node[fill=white] {c} +(0pt,1cm);
\draw[->] (x.60) -- node[fill=white] {d} +(0pt,1cm);
\end{tikzpicture}

\end{document}

As Claudio Fiandrino has mentioned in his comment, another option is to use the calc library, so the shifts are not absolute, but can be calculated in terms of anchors:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}
\node[draw,minimum size=2cm] (x) {X};
\draw[->] ([yshift=-10pt]x.west) -- node[fill=white] {a} +(-1cm,0pt);
\draw[->] ([yshift=10pt]x.west) -- node[fill=white] {b} +(-1cm,0pt);
\draw[->] (x.120) -- node[fill=white] {c} +(0pt,1cm);
\draw[->] (x.60) -- node[fill=white] {d} +(0pt,1cm);
\draw[->]
  ( $ (x.north east)!0.5!(x.east) $ ) -- 
    node[fill=white] {e} 
  +(1cm,0pt);
\draw[->] 
  ( $ (x.east)!0.5!(x.south east) $ ) -- 
    node[fill=white] {f} 
    +(1cm,0pt);
\end{tikzpicture}

\end{document}

In the above example ( $ (x.north east)!0.5!(x.east) $ ) means the point whose coordinate is halfway between x.north east and x.east.

A PSTricks solution:

\documentclass{article}

\usepackage{pstricks-add}
\usepackage{xfp}

\newcommand*\Width{\fpeval{2*\arrowLength+\boxLength}}
\newcommand*\Height{\boxLength}

\def\arrowLength{3}
\def\boxLength{3}


\begin{document}

\begin{pspicture}(\Width,\Height)
 \psset{arrows = ->}
  \psframe(\arrowLength,0)(\fpeval{\arrowLength+\boxLength},\boxLength)
  \rput(\fpeval{\arrowLength+0.5*\boxLength},\fpeval{0.5*\boxLength}){X}
  \pcline(\arrowLength,\fpeval{\boxLength/3})(0,\fpeval{\boxLength/3})
  \ncput*{a}
  \pcline(\arrowLength,\fpeval{2/3*\boxLength})(0,\fpeval{2/3*\boxLength})
  \ncput*{b}
  \pcline(\fpeval{2*\arrowLength+\boxLength},\fpeval{0.5*\boxLength})%
         (\fpeval{\arrowLength+\boxLength},\fpeval{0.5*\boxLength})
  \ncput*{c}
\end{pspicture}

\end{document}

Note that the drawing is 'automated' and all you have to do is choose the values of \arrowLength and \boxLength.