By far the most elegant way to do it is using the siunitx
package:
\documentclass{article}
\usepackage{siunitx}
\sisetup{output-exponent-marker=\textsc{e}}
\begin{document}
\num{6.02e23}
\end{document}

EDIT 1: If you want a short minus sign, a not so elegant solution (but still within siunitx
) is to replace this more complicated sisetup
line:
\sisetup{output-exponent-marker=\textsc{e}, bracket-negative-numbers, open-bracket={\text{-}}, close-bracket={}}

This will in general affect also negative numbers (even without exponents) used within \num
. For other hacks regarding the minus sign look at this question Making a shorter minus or Typesetting of negative versus minus?, no answer on this was really satisfactory to me.
EDIT 2 (2014): If you make XeLaTeX use Écran Font (for example) and some TikZ shadows you can give it a complete look

(note the use of the unicode chacaracter "ᴇ", I needed to use this particular font because it was the only one with this character)
\documentclass{article}
\usepackage{xcolor}
\usepackage{siunitx}
\usepackage{fontspec}
\setmainfont[ExternalLocation={./}]{ecran-monochrome---monochrome-display.ttf} % Écran Monochrome
\usepackage{tikz}
\newcommand\calcshadow[1]{
\tikz[baseline]{
\node[black!50!white] at (0.02,-0.02) {\num[output-exponent-marker=\text{ᴇ}, output-decimal-marker=\text{.}, bracket-negative-numbers, open-bracket={\text{-}}, close-bracket={}]{#1}};
\node[] at (0.,0.) {\num[output-exponent-marker=\text{ᴇ}, output-decimal-marker=\text{.}, bracket-negative-numbers, open-bracket={\text{-}}, close-bracket={}]{#1}};
}
}
\begin{document}
\colorbox{gray!70}{
\calcshadow{6.02e-23}
}
\end{document}
EDIT 3: Alternative style using unicode block character "█":

...
\newcommand\calcshadow[1]{
\tikz[baseline]{
\node[black!20!white] at (0.,0.) {████████};
\node[black!80!white,opacity=0.5] at (0.02,-0.02) {\num[output-exponent-marker=\text{ᴇ}, output-decimal-marker=\text{.}, bracket-negative-numbers, open-bracket={\text{-}}, close-bracket={}]{#1}};
\node[] at (0.,0.) {\num[output-exponent-marker=\text{ᴇ}, output-decimal-marker=\text{.}, bracket-negative-numbers, open-bracket={\text{-}}, close-bracket={}]{#1}};
}
}
...
$6.22\mbox{\textsc{e}}-21$
. I'm assuming this number will be typeset on math mode, right? If you don't want a (typographically correct) math minus sign but a simple dash, be sure to include it in the argument of the\mbox
command.$6.22\mbox{\sc{e}-}21$
is definitely the correct answer, it looks exactly how I'd like it to! You should post that so I can checkmark it.