pgfplots: axis equal at different scales

If what you want is that the division on the x-axis (that are 13) have the same size as the ones of the y-axis (8), you can do the following:

1. remove automatic scaling of the axis, like enlarge y limits
2. use scale only axis, which will apply width and length just to the axis
3. choose a vertical and a horizontal width proportional to the number of divisions.

documentclass[border=5pt, tikz]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=newest}
usepackage{siunitx}
sisetup{per-mode=fraction,
%fraction-function=tfrac
}

begin{document}
begin{tikzpicture}[font=footnotesize,
]
begin{axis}[
xlabel={$t, [si{hour}]$},
ylabel={$displaystyle f(t), left[   si{metrecubedperhour} right]$},
title={$f(t)=frac14 t^3 -12t^2 +144t +250$},
axis lines=middle, % so kein Ursprung
axis x line=bottom,  % so ein Ursprung
scale only axis, 
% 6.15cm = 10cm * hor_division * vert_divisions =10cm/13*8 
width=10cm, height=6.15cm,
xmin=0, xmax=26,
ymin=0, ymax=800,
xtick={0,2,...,26},
ytick={100,200,...,800},
xlabel style={anchor= south east},
ylabel style={anchor= north west},
grid=major, 
]
addplot[thick, smooth, samples=222, domain=0:25, variable=t] {0.25*t^3-12*t^2+144*t+250};
end{axis}
end{tikzpicture}
end{document}

I solved it this way:

pgfmathsetlengthmacrou{6.5mm}

and then

x=0.5*u, y=0.01*u

documentclass[border=5pt, tikz]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=newest}
usepackage{siunitx}
sisetup{per-mode=fraction,
%fraction-function=tfrac
}

pgfmathsetlengthmacrou{6.5mm}

begin{document}
begin{tikzpicture}[font=footnotesize,
]
begin{axis}[
xlabel={$t, [si{hour}]$},
ylabel={$displaystyle f(t), left[   si{metrecubedperhour} right]$},
title={$f(t)=frac14 t^3 -12t^2 +144t +250$},
axis lines=middle, % so kein Ursprung
axis x line=bottom,  % so ein Ursprung
x=0.5*u, y=0.01*u,
xmin=0, xmax=25.5,
ymin=0, ymax=800,
xtick={0,2,...,24},
ytick={100,200,...,800},
xlabel style={anchor= south east},
ylabel style={anchor= north west},
grid=major, 
%enlarge x limits={abs=1.1, upper},
enlarge y limits={rel=0.1, upper},
]
addplot[thick, smooth, samples=222, domain=0:25, variable=t] {0.25*t^3-12*t^2+144*t+250};
end{axis}
end{tikzpicture}
end{document}