How to place and number 3 short equations in one line?

First of all, Avoid eqnarray.

You can use a normal equation environment and separate the equations with a \qquad.

\documentclass[11pt, a4paper]{scrbook}

\usepackage[ngerman]{babel}
\usepackage{amsmath}

\begin{document}

\begin{equation}
x_{1} = \frac{A-A_{0}}{\frac{1}{2}\cdot \left( A_{\mathrm{A}} - A_{\mathrm{a}} \right)}
\qquad
x_{2} = \frac{B-B_{0}}{\frac{1}{2}\cdot \left( B_{\mathrm{A}} - B_{\mathrm{a}} \right)}
\qquad
x_{3} =  \frac{C-C_{0}}{\frac{1}{2}\cdot \left( C_{\mathrm{A}} - C_{\mathrm{a}} \right)}
\label{exv:eqn:UmrechnungEingangsgroesse3}
\end{equation}
\end{document}

(In my original answer I used an align environment, but this isn't really "correct", as that is meant for aligning equations over more than one line. As this only has one line, it is more appropriate to use an equation environment, as commented by Andrew Stacey.)

Firstly, I fully support the points avoiding eqnarray and using amsmath environments whenever possible, such as align.

amsmath provides a subequations environment, which might also be useful, though it's more for numbering subequations but not for lining up horizontally.

You could use align such as Torbjørn showed, as it supports several columns. Here the & symbol is both for alignment and for separating columns, as in tabular, alternating.

Another option is flalign, you could use it in the very same way, but the equations would be more spread out, so you would have more space.

You don't need to use a amsmath align environment, since the horizontal positioning is a tabular like issue. So if you would like to have equations with labels you can refer to, you could use for example tabularx with equal column width:

\documentclass[11pt,a4paper]{scrbook}
\usepackage{amsmath}
\usepackage{tabularx}
\begin{document}
\chapter{Equations}
See equations \eqref{eqn:1}, \eqref{eqn:2} and \eqref{eqn:3}.

\noindent\begin{tabularx}{\textwidth}{@{}XXX@{}}
  \begin{equation}
  x_{1} = \frac{A-A_{0}}{\frac{1}{2}
    \cdot \left( A_{\mathrm{A}} - A_{\mathrm{a}} \right)}
    \label{eqn:1}
  \end{equation} &
  \begin{equation}
  x_{2} = \frac{B-B_{0}}{\frac{1}{2}
    \cdot \left( B_{\mathrm{A}} - B_{\mathrm{a}} \right)}
    \label{eqn:2}
  \end{equation} &
  \begin{equation}
  x_{3} = \frac{C-C_{0}}{\frac{1}{2}
    \cdot \left( C_{\mathrm{A}} - C_{\mathrm{a}} \right)}
    \label{eqn:3}
  \end{equation}
\end{tabularx}
\end{document}