How to improve TikZ diagram with a lot of edges?

like this?

edit

in comparison to your code the code in mwe, by which the above image was drawm, has the following changes/improvements:

• by introducing library quotes writing of edge labels is simplified and much shorter. for example, instead of

\draw (H) edge[bend right=10,geldstrom] node[left] {3} (B);

is possible obtain the same result with

\draw (H) edge ["3"] (B)

where for edge i define style, which determine edge color, thickness, shortness and bending angle:

every edge/.style = {draw=SlateGrey, -Stealth, line width=2pt,
                     shorten <=5pt, shorten > = 5pt,
                     bend right=8},

• the angle of bending is essentially determined by trial and error method. however there i exploit my experiences from drawing of the similar diagrams and knowledge of geometry (already my first trial was successful). in the edge style i determine the most common angle, exceptions for four edges are determined locally.
• since edges style is defined, there is no more need to write \draw ... : for each edge. it is sufficient write it ones and then list all edges (see mwe below).
• with style edge quotes is determined font size and position of edge labels.
• size of nodes shapes is also changed. used bigger height (and also with shortening of edges) prevent edges overlaps.
• with library arrows.meta is defined on decent ways arrows heads

• summary of tikzpicture changes are:

  • style subjekt is renewed
  • styles strom, gueterstrom and geldstrom are replaced by every edge and every edge quotes

\documentclass[parskip=half]{scrreprt}
\usepackage[svgnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,
                positioning,
                quotes,
                shapes}

\begin{document}
  \begin{tikzpicture}[auto=right,
    node distance = 44mm,
   subjekt/.style = {draw, semithick, minimum height=12mm, minimum width=24mm},
every edge/.style = {draw=SlateGrey, -Stealth, line width=2pt,
                     shorten <=5pt, shorten > = 5pt,
                     bend right=8},
every edge quotes/.append style = {inner sep=2pt, font=\small, pos=0.6},
                    ]
\node (H) [subjekt]                     {Haushalt};
\node (U) [subjekt,below right=of H]    {Unternehmen};
\node (S) [subjekt,above right=of U]    {Staat};
\node (B) [subjekt,below left =of U]    {Banken};
\node (A) [subjekt,below right=of U]    {Ausland};
%
\draw   (H) edge ["3"]  (B)
        (B) edge ["9"]  (H)
        (H) edge ["1"]  (S)
        (S) edge ["13"] (H)
        (H) edge ["2"]  (U)
        (U) edge ["6"]  (H)
%
        (H) edge [bend right=24, "4"]  (A)
        (A) edge [bend right=24, "20"] (H)
%
        (B) edge ["10"] (U)
        (U) edge ["7"]  (B)
        (B) edge ["12"] (A)
        (A) edge ["19"] (B)
%
        (B) edge [bend right=24, "11"] (S)
        (S) edge [bend right=24, "16"] (B)
%
        (A) edge ["18"] (U)
        (U) edge ["8"]  (A)
        (A) edge ["17"] (S)
        (S) edge ["14"] (A)
        (S) edge ["15"] (U)
        (U) edge ["5"]  (S);
  \end{tikzpicture}
\end{document}

The arrangement you chose is very suboptimal for a fully connected graph. It is better to arrange things in a circular layout.

TikZ graphdrawing requires LuaTeX.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{graphs,graphdrawing,quotes}
\usegdlibrary{circular}
\begin{document}

\begin{tikzpicture}
  \graph[
  simple necklace layout,
  edges={bend right=10},
  edge quotes={fill=white,inner sep=1pt,font=\tiny},
  nodes={draw},
  node sep=5em,
  ] {
    Haushalt -> {Staat [>"1"], Unternehmen [>"2"], Banken [>"3"], Ausland [>"4"]},
    Unternehmen -> {Haushalt [>"6"], Staat [>"5"], Banken [>"7"], Ausland [>"8"]},
    Banken -> {Haushalt [>"9"], Staat [>"11"], Unternehmen [>"10"], Ausland [>"12"]},
    Staat -> {Haushalt [>"13"], Unternehmen [>"15"], Banken [>"16"], Ausland [>"14"]},
    Ausland -> {Haushalt [>"20"], Staat [>"17"], Unternehmen [>"18"], Banken [>"19"]}
  };
\end{tikzpicture}

\end{document}

How to improve the situation in the corners? Get rid of the corners! I also changed the arrow tips and tightened things up a bit.

\documentclass[parskip=half]{scrreprt}
\usepackage[svgnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{positioning,shapes}
\tikzset{subjekt/.style={circle,draw,minimum height=.7cm, minimum width=2.2cm, align=center},
         strom/.style={line width=2pt, shorten >=2pt,shorten <=2pt},
         gueterstrom/.style={strom,Goldenrod,->},
         geldstrom/.style={strom,SlateGrey,->}}
\begin{document}
  \begin{tikzpicture}[node distance=4cm,>=stealth]
    \node[subjekt] (H) at (0,0) {Haushalt};
    \node[subjekt] (U) [below right= of H] {Unternehmen};
    \node[subjekt] (S) [above right= of U] {Staat};
    \node[subjekt] (B) [below left= of U]  {Banken};
    \node[subjekt] (A) [below right= of U] {Ausland};
    \draw (H) edge[bend right=10,geldstrom] node[left] {3} (B);
    \draw (B) edge[bend right=10,geldstrom] node[right] {9} (H);
    \draw (H) edge[bend right=10,geldstrom] node[below] {1} (S);
    \draw (S) edge[bend right=10,geldstrom] node[above] {13} (H);
    \draw (H) edge[bend right=10,geldstrom] node[below left] {2} (U);
    \draw (U) edge[bend right=10,geldstrom] node[above right] {6} (H);
    \draw (H) edge[bend right=22,geldstrom, near end] node[below left] {4} (A);
    \draw (A) edge[bend right=22,geldstrom, near end] node[above right] {20} (H);
    \draw (B) edge[bend right=10,geldstrom] node[below right] {10} (U);
    \draw (U) edge[bend right=10,geldstrom] node[above left] {7} (B);
    \draw (B) edge[bend right=10,geldstrom] node[below] {12} (A);
    \draw (A) edge[bend right=10,geldstrom] node[above] {19} (B);
    \draw (B) edge[bend right=22,geldstrom, near end] node[below right] {11} (S);
    \draw (S) edge[bend right=22,geldstrom, near end] node[above left] {16} (B);
    \draw (A) edge[bend right=10,geldstrom] node[above right] {18} (U);
    \draw (U) edge[bend right=10,geldstrom] node[below left] {8} (A);
    \draw (A) edge[bend right=10,geldstrom] node[right] {17} (S);
    \draw (S) edge[bend right=10,geldstrom] node[left] {14} (A);
    \draw (S) edge[bend right=10,geldstrom] node[above left] {15} (U);
    \draw (U) edge[bend right=10,geldstrom] node[below right] {5} (S);
  \end{tikzpicture}
\end{document}