The arrow bodies are here, but tiny. Use the pgfplots/quiver/arrows scale
key to make them bigger.
Unfortunately, your data is uneasy to plot as a quiver, due to strong variations of the values.
Maybe you should scale them non-linearly (e.g. replace the norm of the arrows by some logarithm of their norm)
Also some good (maybe better) suggestions are given in comments.
See also the second version, below, that looks better.
\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[title=Quiver and plot table]
\addplot
[
blue,
quiver=
{
u=\thisrow{u},
v=\thisrow{v},
scale arrows=200, %to be adjusted
},
-stealth
]
table {vector_field.dat};
\end{axis}
\end{tikzpicture}
\end{document}
Another version, taken from the pgfplots manual.
\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\begin{document}
% define some constants:
\def\U{\thisrow{u}}
\def\V{\thisrow{v}}
\def\LEN{(sqrt((\U)^2 + (\V)^2)}
\begin{tikzpicture}
\begin{axis}[title=Quiver and plot table]
\addplot
[
blue,
point meta={\LEN},
quiver=
{
u={(\U)/\LEN}, v={(\V)/\LEN},
scale arrows=.2,
every arrow/.append style=
{
line width=2pt*\pgfplotspointmetatransformed/1000
},
},
-stealth
]
table {vector_field.dat};
\end{axis}
\end{tikzpicture}
\end{document}
yielding :
Cheers,