Use \singlespacing in scrheading and \onehalfspacing in the main document

First, prepend \setstretch{1} to the argument of \chead. (Don't use \singlespacing because that would add extra space at the start.) Second, because a singlespaced line's height is 0.8 times the height of a onehalfspaced line, change headlines=3 to headlines=2.4.

\documentclass[fontsize=14pt,headlines=2.4,headinclude]{scrartcl}
\usepackage[onehalfspacing]{setspace}
\usepackage{lipsum}
\usepackage{scrpage2}
\chead{\setstretch{1}Some really long text\\ that takes\\more than one line}
\pagestyle{scrheadings}
\begin{document}
\lipsum
\end{document}

I'd use a \parbox; but since scrpage2 already uses a \parbox itself for placing the header, the final result will be almost unpredictable, unless we do the trick of hiding the height of the box

\documentclass[fontsize=14pt,headlines=3,headinclude]{scrartcl}
\usepackage[pass,showframe]{geometry} % just to draw frames
\usepackage{fontspec}
\setmainfont{CMU Serif}
\usepackage{setspace}
\usepackage{lipsum}
\usepackage{scrpage2}
\chead{%
  \parbox[b][0pt]{\textwidth}{
  \singlespacing\centering
    Some really long text\\
    that takes\\
    more than one line\vspace{-\dp\strutbox}
  }%
}

\pagestyle{scrheadings}

\begin{document}
\onehalfspacing
\lipsum
\end{document}

The final \vspace is meant to push the bottom line in the “really correct” position (a couple of points, actually).

Since 2013 scrpage2 is obsolete. Since version 3.24 the official successor scrlayer-scrpage has an option singlespacing to set the page head and footer in singlespacing:

\documentclass[fontsize=14pt,headlines=3,headinclude]{scrartcl}
\usepackage{setspace}
\usepackage{lipsum}
\usepackage[singlespacing]{scrlayer-scrpage}
\chead{Some really long text\\ that takes\\more than one line}
\pagestyle{scrheadings}

\begin{document}
\onehalfspacing
\lipsum
\end{document}

results in: