Place a list in the middle of the page

Without guessing:

\documentclass{article}
\usepackage{environ}
\usepackage{lipsum} % just for the example

\NewEnviron{centeritemize}{%
  \global\centeritemizewd=0pt
  \setbox0=\vbox{
    \renewcommand{\item}[1][]{\centeritemizeitem}
    \BODY\item\centeritemize
  }
  \global\advance\centeritemizewd\labelwidth
  \begin{center}
  \begin{minipage}{\centeritemizewd}
  \leftmargini=0pt
  \begin{itemize}
  \BODY
  \end{itemize}
  \end{minipage}
  \end{center}
}
\newdimen\centeritemizewd
\def\centeritemizeitem#1\item#2{%
  \setbox2=\hbox{#1}%
  \ifdim\wd2>\centeritemizewd
    \global\centeritemizewd=\wd2
  \fi
  \ifx#2\centeritemize
    % end the recursion
  \else
    \expandafter\centeritemizeitem
  \fi
}

\begin{document}
\lipsum*[2]
\begin{centeritemize}
  \item First item
  \item Second item
\end{centeritemize}
\lipsum[2]
\end{document}

Limitations: no blank lines are allowed in the centeritemize environment.

Don't use it. There's no reason to.

\begin{center}
\begin{minipage}{.5\textwidth}
\begin{itemize}

    \item First item

    \item Second item

\end{itemize}
\end{minipage}
\end{center}

EDIT: Most methods try to guess the width of the \itemize by trial and error. Here is another simple method that requires no guessing (making use of the powerfull varwidth package):

\documentclass{article}
\setlength{\parindent}{0in}
\usepackage{varwidth}
\begin{document}

\newlength\mylen
\newsavebox\mybox
\savebox\mybox{\begin{varwidth}{\textwidth}\textbullet\qquad Second item\end{varwidth}}
\setlength\mylen{\the\wd\mybox}

\vspace*{\fill}\centering
\parbox{\mylen}{%
\begin{itemize}
    \item First item
    \item Second item
\end{itemize}}
\vspace*{\fill}

\end{document}

OLD sol: Can try this (without changing the whole page size):

\documentclass{article}
\setlength{\parindent}{0in}
\begin{document}

\vspace*{\fill}\centering
\parbox{3cm}{%
\begin{itemize}
    \item First item
    \item Second item
\end{itemize}}
\vspace*{\fill}
\end{document}

Giving:

Limitation here is guessing the 3cm width.