How can I number paragraphs without higher level counters?

By default paragraphs are numbered within subsections, restarting at 1 at each subsection. If you just wish to changed the printed representation of the paragraph number then all you need is

\renewcommand{\theparagraph}{\S\arabic{paragraph}}
\setcounter{secnumdepth}{4}

If you wish the paragraphs to also be numbered consecutively throughout the document then you can use the chngcntr package's command \counterwithout to undo the resetting as follows:

\documentclass[12pt]{article}

\usepackage{chngcntr}
\counterwithout{paragraph}{subsubsection}
\renewcommand{\theparagraph}{\S\arabic{paragraph}}
\setcounter{secnumdepth}{4}

\begin{document}

\section{First section}

\paragraph{}\label{Personal Statement}
San Francisco is a city in California.

\subsection{A subsection}

\paragraph{}
San Francisco is a city in California.  See paragraph \ref{Personal
Statement}.

\subsubsection{A subsubsection}

\paragraph{}
San Francisco is a city in California.

\section{Second section}

\paragraph{}
San Francisco is a city in California. See paragraph \ref{Personal Statement}.

\paragraph{}
San Francisco is a city in California.


\end{document}

Added in response to comment To adjust the spacing after numbers etc. it is probably easiest to use the titlesec package. There are two commands involved here: \titleformat and \titlespacing. If you never intend to give titles to your paragraphs then the \titlespacing command below is sufficient. The \titleformat command makes sure words in a paragraph heading are also handled appropriately:

\documentclass[12pt]{article}

\usepackage{chngcntr}
\counterwithout{paragraph}{subsubsection}
\renewcommand{\theparagraph}{\S\arabic{paragraph}}
\setcounter{secnumdepth}{4}

\usepackage{titlesec}
\titleformat{\paragraph}[runin]{\normalfont\bfseries}{\theparagraph}{\wordsep}{}
\titlespacing{\paragraph}{0pt}{3.25ex plus 1ex minus .2ex}{\wordsep}

\begin{document}

\section{First section}

\paragraph{}\label{Personal Statement}
San Francisco is a city in California.

\subsection{A subsection}

\paragraph{}
San Francisco is a city in California.  See paragraph \ref{Personal
Statement}.

\subsubsection{A subsubsection}

\paragraph{}
San Francisco is a city in California.

\section{Second section}

\paragraph{}
San Francisco is a city in California. See paragraph \ref{Personal Statement}.

\paragraph{With title}
San Francisco is a city in California.

\end{document}

The arguments given to \titlespacing are as follows:

• section command = \paragraph
• left indent = 0pt
• above vertical skip = 3.25ex plus 1ex minus .2ex the standard value from the article class
• final separation = \wordsep an ordinary word space

For \titleformat the spcification is

• section command = \paragraph
• style = runin for heads contain in the same paragraph as the text
• format = \normalfont\bfseries for styling the heading
• label = \theparagraph the printed representation of the section number
• sep = \wordsep is the space between the label and the title (when present)
• before code = empty

You can use package scrjura of KOMA-Script.

An example could be (as @harish requested in his comment ;-):

\documentclass[%
  ngerman
 ,paper=a4
 ,fontsize=12pt
 ,parskip=half-
 ,pagesize
 ,numbers=noenddot
]{scrartcl}% scrreprt scrartcl

\usepackage{babel}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[%
  juratotoc%=1              % juratotoc=0 0 chapter, 1 section
 ,paragraphmark=forceboth
 ,juratocnumberwidth=2.5em
]{scrjura}  

\usepackage{enumerate}

\makeatletter
% to get rid of a warning ...
\providecommand*{\toclevel@cpar}{0}
\makeatother


\begin{document}

\subject{Subject}
\title{Title}
\subtitle{Subtitle}
\author{}   % no author: {} empty
%\date{}    % no date

\maketitle  % setzt den Titel der Satzung

\addsec{Preface}
A preface, if you need it.


%\clearpage 
\appendix % Letters instead of numbers

\section{Section One}
\begin{contract}
\Paragraph{title={Title first paragraph}}
The content of first paragraph.

A list of posibilities:
\begin{enumerate}[\qquad a)]
  \item first.
  \item second.
  \item third.
\end{enumerate}

The content of third paragraph.
\end{contract}

\section{Section Two}
\begin{contract}
\Paragraph{title={Title first paragraph}}
The content of first paragraph.
\end{contract}

\end{document} 

With the result: