How would you produce the following exact hexagon?

I'd use TikZ to draw that diagram

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[>=latex]
    \def\radius{2cm} % change to an appropriate value
    \node (h0A) at (60:\radius)   {$H^0(A)$};
    \node (h0C) at (0:\radius)    {$H^0(C)$};
    \node (h1B) at (-60:\radius)  {$H^1(B)$};
    \node (h1A) at (-120:\radius) {$H^1(A)$};
    \node (h1C) at (180:\radius)  {$H^1(C)$};
    \node (h0B) at (120:\radius)  {$H^0(B)$};

    \path[->,font=\small]
        (h0A) edge node[auto] {$g_0$} (h0C)
        (h0C) edge node[auto] {$\delta_0$} (h1B)
        (h1B) edge node[auto] {$f_1$} (h1A)
        (h1A) edge node[auto] {$g_1$} (h1C)
        (h1C) edge node[auto] {$\delta_1$} (h0B)
        (h0B) edge node[auto] {$f_0$} (h0A);
\end{tikzpicture}
\end{document}

First the 6 vertices are defined (as nodes in TikZ-speak) using polar coordinates. They are named (h0A) to (h0B). Then the arrows are drawn using edge paths ((start) edge (finish)) and the descriptions (again nodes) are placed beside them with node[auto] {text}.

A nice introduction to drawing commutative diagrams with TikZ is Commutative Diagrams with TikZ.

Here's a way with Xy-pic

\documentclass[a4paper]{article}
\usepackage[all,cmtip,pdf]{xy}
\usepackage{amsmath}
\begin{document}

\begin{equation}
\begin{gathered}
\xymatrix@C-1em{
& H^0(B) \ar[rr]^{f_0} && H^0(A) \ar[dr]^{g_0} \\
H^1(C) \ar[ur]^{\delta_1} &&&& H^0(C) \ar[dl]^{\delta_0} \\
& H^1(A) \ar[ul]^{g_1} && H^1(B) \ar[ll]^{f_1}
}
\end{gathered}
\end{equation}
\end{document}

Remember to put the \xymatrix inside a gathered environment if you need an equation number next to it. Otherwise discard the gathered environment.

I've reduced the column distance and doubled the middle arrow across an empty column to keep uniform the arrow lengths.

Run with xelatex

\documentclass{article}
\usepackage{pst-node}

\begin{document}

\def\R{2.5}
\begin{pspicture}(-3,-3)(3,3)
\psframe*[linecolor=red!10](-3,-3)(3,3)
\psset{nodesep=3pt,arrows=->,shortput=nab}\degrees[6]
\psnode(\R;0){P0}{$H^0(C)$} \psnode(\R;1){P1}{$H^0(B)$}
\psnode(\R;2){P2}{$H^0(B)$} \psnode(\R;3){P3}{$H^1(C)$}
\psnode(\R;4){P4}{$H^1(A)$} \psnode(\R;5){P5}{$H^1(B)$}
\ncline{P0}{P5}^{$\delta_0$}\ncline{P5}{P4}^{$f_1$}
\ncline{P4}{P3}^{$g_1$}     \ncline{P3}{P2}^{$\delta_1$}
\ncline{P2}{P1}^{$f_0$}     \ncline{P1}{P0}^{$g_0$}
\end{pspicture}

\end{document}