You can define a customized math "function" which relies on the fact that it is evaluated while the input table is read. In other words, it is only valid within your context.
It uses the FPU of pgf which is used by pgfplots, and, in particular, its method \pgfmathfloatparsenumber. A float in pgf has "flags" which is an integer with the meaning
0 == '0' (the number is +- 0.0),
1 == '+',
2 == '-',
3 == 'not a number'
4 == '+ infinity'
5 == '- infinity'
consequently, we can call \pgfmathfloatparsenumber and use \pgfmathfloatgetflagstomacro to access the flag. Integer comparisons can be done by means of \ifnum<int>=<int> \else \fi.
Defining two functions, one for Y and one for the error bar results in
\documentclass[11pt]{article}
\usepackage{pgfplots}
\usepackage{filecontents}
\usepackage{verbatim}
\begin{filecontents*}{data.dat}
time w1 e1 w2 e2
1 3019 40 nan nan
2 3045 34 nan nan
3 3100 50 3104 24
4 3500 13 3498 90
5 3800 90 3803 12
6 nan nan 3980 43
7 nan nan 3985 80
\end{filecontents*}
\begin{document}
\thispagestyle{empty}
% if (index 1 and index 3 are not NaN):
% y expr= (index 1 + index 3)/2
% y error expr= max{index 2, index 4}
% else if (index 3 = NaN): % && index 1 is not NaN
% y index =1
% y error index=2
% else if (index 1 = NaN): % && index 3 is not NaN
% y index =3
% y error index=4
\pgfmathdeclarefunction{VAL}{0}{%
\pgfmathfloatparsenumber{\thisrowno{1}}\let\A=\pgfmathresult
\pgfmathfloatparsenumber{\thisrowno{3}}\let\C=\pgfmathresult
\pgfmathfloatgetflagstomacro\A\flags
\ifnum\flags=3 %
\pgfmathfloatgetflagstomacro\C\flags
\ifnum\flags=3 %
% A == nan && C == nan
\let\pgfmathresult=\A
\else
% A == nan && C != nan
\let\pgfmathresult=\C
\fi
\else
\pgfmathfloatgetflagstomacro\C\flags
\ifnum\flags=3 %
% A != nan && C == nan
\let\pgfmathresult=\A
\else
% A != nan && C != nan
\pgfmathparse{(\A + \C)/2}%
\fi
\fi
}%
\pgfmathdeclarefunction{VALERR}{0}{%
\pgfmathfloatparsenumber{\thisrowno{1}}\let\A=\pgfmathresult
\pgfmathfloatparsenumber{\thisrowno{2}}\let\B=\pgfmathresult
\pgfmathfloatparsenumber{\thisrowno{3}}\let\C=\pgfmathresult
\pgfmathfloatparsenumber{\thisrowno{4}}\let\D=\pgfmathresult
\pgfmathfloatgetflagstomacro\A\flags
\ifnum\flags=3 %
\pgfmathfloatgetflagstomacro\C\flags
\ifnum\flags=3 %
% A == nan && C == nan
\let\pgfmathresult=\B
\else
% A == nan && C != nan
\let\pgfmathresult=\D
\fi
\else
\pgfmathfloatgetflagstomacro\C\flags
\ifnum\flags=3 %
% A != nan && C == nan
\let\pgfmathresult=\B
\else
% A != nan && C != nan
\pgfmathparse{max(\B,\D)}%
\fi
\fi
}%
\begin{tikzpicture}
\begin{axis} [
compat=newest,
y tick label style={/pgf/number format/1000 sep=},
ymin=2900,
ymax=4400]
\addplot [color=blue, thick,
error bars/.cd,
y explicit,
y dir=both,
] table [
x index=0,
y expr=VAL,
y error expr=VALERR,
]
{data.dat};
\addlegendentry{fit}
\end{axis}
\end{tikzpicture}
\end{document}
note that I have eliminated the two other \addplot statements.
The \let\<macro>=<othermacro> copies <othermacro> to <macro>. The constructions \ifnum<\macro>=3 % compares two integers. Note the trailing space after 3, it is important to tell TeX that it should stop to search for further numeric literals. It will be omitted from the output.
Best Answer
As you noted yourself, using
nanas a placeholder for empty values solves the problem. It's worth pointing out that you also have the choice of whether to just skip the empty value and connect the surrounding data points, or whether the graph should be interrupted when it encounters an empty value. You can control this behaviour usingunbounded coords=discard(which is the default behaviour) orunbounded coords=jump.