Use \captionof in conjunction with \subfigure?

Since this seems like an out-of-the-ordinary setting for a figure, it may require some manual fiddling. The following example sets a list-like environment containing the "figure", and it allows your caption to break across the page boundary.

\documentclass{scrartcl}
\usepackage{graphicx,lipsum,afterpage,subcaption}
\usepackage{enumitem}
\begin{document}

\lipsum[1-5]

\makeatletter
\afterpage{\begin{center}
    \def\@captype{figure}% Fake a figure environment
    \subcaptionbox{Sub1}{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subcaptionbox{Sub2}{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subcaptionbox{Sub3}{\includegraphics[width=.2\linewidth]{example-image-a}} \par\bigskip
    \subcaptionbox{Sub4}{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subcaptionbox{Sub5}{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subcaptionbox{Sub6}{\includegraphics[width=.2\linewidth]{example-image-a}}
  \end{center}
  \addcontentsline{lof}{figure}{\protect\numberline{\thefigure}A length figure}% LoF entry
  \settowidth{\@tempdima}{Figure~\thefigure: }%
  \begin{itemize}[labelindent=\parindent,labelwidth=\@tempdima,rightmargin=\parindent,leftmargin=\dimexpr\@tempdima+\parindent]
    \item[Figure~\thefigure:]
    \lipsum[6-10]
  \end{itemize}
}
\makeatother

\lipsum[6-12]

\end{document}

The sub-figures are set using subcaption using the regular interface inside a center environment. A faked figure is initiated by defining the caption type \@captype. Setting of the caption is inside a regular list itemize with appropriate width settings (to match your other figures).

A short LoF-entry is set manually via \addcontentsline{lof}{figure}{<short entry>}. This is displayed with the inclusion of \listoffigures.

All of the above is passed to \afterpage (supplied by afterpage) in order to provide a seemless transition from page-to-page. One may play around with the spacing following the "figure". For example, adding something like \addvspace{\textfloatsep}.

_{The page layout was added via \usepackage{showframe} in the preamble; removed in the above code.}

This solution is based on three ideas:

1. We use a center environment instead of a figure since the content of a figure is limited to one page only. (This idea was stolen from Werner.) We use \setcaptiontype{figure} (or \captionsetup{type=figure}) to make this environment known to the (sub)caption package as figure. BTW: Please don't use \def\@captype{figure} when using the caption package. When doing so this solution won't work correctly with hyperref, caption settings for figures would not apply etc.
2. We use \afterpage to let this thing start on a page. (This idea was stolen from Werner, too.)
3. We use \captionsetup{box=none,parbox=none} to avoid the usage of boxes when typesetting the caption since boxes would limit the caption to one page only. (Finally a idea from myself ;-))

Example code, shamelessly stolen from Werner and modified:

\documentclass{scrartcl}
\usepackage{graphicx,lipsum,afterpage,subcaption}
\begin{document}

\lipsum[1-5]

\afterpage{\begin{center}
    \setcaptiontype{figure}% Fake a figure environment
    \subcaptionbox{Sub1}{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subcaptionbox{Sub2}{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subcaptionbox{Sub3}{\includegraphics[width=.2\linewidth]{example-image-a}} \par\bigskip
    \subcaptionbox{Sub4}{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subcaptionbox{Sub5}{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subcaptionbox{Sub6}{\includegraphics[width=.2\linewidth]{example-image-a}}
    \captionsetup{box=none,parbox=none}
    \caption[A length figure]{%
      \lipsum[6-10]}
  \end{center}
}

\lipsum[6-12]

\end{document}

There is no need to switch to the subcaption package since this solution works with the subfigure package, too:

\documentclass{scrartcl}
\usepackage{graphicx,lipsum,afterpage,caption,subfigure}
\begin{document}

\lipsum[1-5]

\afterpage{\begin{center}
    \setcaptiontype{figure}% Fake a figure environment
    \subfigure[Sub1]{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subfigure[Sub2]{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subfigure[Sub3]{\includegraphics[width=.2\linewidth]{example-image-a}} \par\bigskip
    \subfigure[Sub4]{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subfigure[Sub5]{\includegraphics[width=.2\linewidth]{example-image-a}} \qquad
    \subfigure[Sub6]{\includegraphics[width=.2\linewidth]{example-image-a}}
    \captionsetup{box=none,parbox=none}
    \caption[A length figure]{%
      \lipsum[6-10]}
  \end{center}
}

\lipsum[6-12]

\end{document}