Section numbering in binary

The fmtcount package provides a simple way to get the binary numbers. To begin counting at zero you need to set the section counter initially to -1. You can also pad the values with zeroes if you wish.

\documentclass{article}
\usepackage{fmtcount}
\renewcommand{\thesection}{\binary{section}.}
\setcounter{section}{-1}% To begin at 0

\begin{document}

\section{First section}
\section{Second section}
\section{Third section}
\section{Fourth section}
\section{Fifth section}
\renewcommand{\thesection}{\padzeroes[4]\binary{section}}
\setcounter{section}{-1}
\section{First section}
\section{Second section}
\section{Third section}
\section{Fourth section}
\section{Fifth section}
\end{document}

Sure.

\documentclass{article}

\usepackage{expl3}

\ExplSyntaxOn
    \cs_set:Npn \thesection {\int_to_bin:n{\value{section}-1}}
\ExplSyntaxOff

\begin{document}

\section{one}

\section{two}

\section{three}

\end{document}

Note that I only changed the displayed number here, so the section counter still has the value 1 for the first section, we only print 0. If you want the section to "really" have the number 0, you need to remove the -1 from the definition of \thesection and set the counter to -1 at the start of the document.