There is a paleo-babylonian font on this page ^[1]. Using fontspec with XeTeX or LuaTex and things like \char"1240D, you could easily typeset what you need.

Here is a quick example for XeTeX.

Note: It turns out that the font doc is wrong (Ah! If they used TeX to generate it...) and 9 is actually mapped at 1240E, quite logically. Also, 20 seems to be missing while 30 is mapped several times for some reason, so I'm doing 20 with 2 "10" glyphs and a bit of kerning. I'm also adjust kerning between tens and units.

\documentclass[paper=a4]{minimal}

\usepackage{fontspec}
\usepackage{xunicode}

\usepackage{pgffor}
\usepackage{numname}

\newcommand{\babydisp}[1]{%
  \fontspec{Santakku}{\char"#1}}

\makeatletter
\newcommand{\babynum}[1]{%
\numdigits{#1}%
\ifcase\c@xsm@mctr
  \or
  \babydisp{1230B} %10
  \or
  \babydisp{1230B}\kern-1.5pt{}\babydisp{1230B} %20 -- unknown?
  \or
  \babydisp{1230D} %30
  \or
  \babydisp{1240F} %40
  \or
  \babydisp{12410} %50
\fi
\kern-4pt{}%
\ifcase\c@ism@mctr
  \or
  \babydisp{12079} %1
  \or
  \babydisp{1222B} %2
  \or
  \babydisp{12408} %3
  \or
  \babydisp{120FB} %4
  \or
  \babydisp{1240A} %5
  \or
  \babydisp{1240B} %6
  \or
  \babydisp{1240C} %7
  \or
  \babydisp{1240D} %8
  \or
  \babydisp{1240E} %9
\fi
}
\makeatother

\begin{document}

%\noindent\foreach \n in {1,...,59}{%
%\n\quad\babynum{\n}\\
%}

\begin{minipage}{0.2\textwidth}
\noindent\foreach \n in {1,...,20}{%
\n\quad\babynum{\n}\\
}
\end{minipage}
\begin{minipage}{0.2\textwidth}
\noindent\foreach \n in {21,...,40}{%
\n\quad\babynum{\n}\\
}
\end{minipage}
\begin{minipage}{0.2\textwidth}
\noindent\foreach \n in {41,...,59}{%
\n\quad\babynum{\n}\\
}
\end{minipage} \hfill

\end{document}

Edit: There is now a package on CTAN ^[2], so you can use it as follows:

\documentclass[paper=a4]{minimal}

\usepackage{pgffor}
\usepackage{babyloniannum}


\begin{document}

\begin{minipage}{0.2\textwidth}
\noindent\foreach \n in {1,...,20}{%
\n\quad\babyloniannum{\n}\\
}
\end{minipage}
\begin{minipage}{0.2\textwidth}
\noindent\foreach \n in {21,...,40}{%
\n\quad\babyloniannum{\n}\\
}
\end{minipage}
\begin{minipage}{0.2\textwidth}
\noindent\foreach \n in {41,...,59}{%
\n\quad\babyloniannum{\n}\\
}
\end{minipage} \hfill

\end{document}

Edit: Version 0.4 of the package allows to typeset numbers beyond 59 (up to 60^9 = 1.0077696 × 10^16 in theory, although I think TeX will give up before that).

Wikipedia also has the two needed symbols as SVG vector graphics:
http://en.wikipedia.org/wiki/File:Babylonian_1.svg
http://en.wikipedia.org/wiki/File:Babylonian_10.svg

You can turn them rather easily to PDF using e.g. inkscape --export-pdf=<pdf> <svg>. Then position them in the shown way for all numbers from 1-59. For larger numbers you would need to add some loop.

It is also possible to make your own font as shown in Create a symbol font from SVG symbols ^[1].