The default section numbering is
\newcommand*{\thesection}{\thechapter.\arabic{section}}
If \thechapter is empty, the dot is still there. So you would have to redefine \thesection too:
\renewcommand*{\thesection}{\arabic{section}}
This will automatically also change \thesubsection … \thesubparagraph, because they all depend directly or indirectly on \thesection.
But what about figures, tables, equations? They all have similar definitions like the section. So you would also need to redefine \thefigure, \thetable and \theequation similar to \thesection. But then you will have several sections, subsection … subparagraphs and several figures, tables and equations with the same number. So I would recommend not just to redefine \thesection, \thetable, \thefigure and \theequation but make their counters independent from the chapter counter. You can use \counterwithout to do so, e.g.,
\counterwithout{section}{chapter}
to make the section counter independent from the chapter counter. This will also redefine \thesection like I've shown with the explicit redefinition above.
But this is not enough. In your example the chapter titles still start with an indent. This is the space between the (not existing) chapter number and the chapter title text. So you have to remove this space too. You can do this redefining \chapterformat:
\renewcommand*{\chapterformat}{}% No number and no space
You should do the same for the running heads (used if you are using \pagestyle{headings}):
\renewcommand*{\chaptermarkformat}{}
All together I would recommend:
\documentclass{scrreprt}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}% Not needed with LaTeX from 2018-04-01
% \usepackage{chngcntr}% Only needed with old LaTeX.
\renewcommand*{\thechapter}{}
\renewcommand*{\chapterformat}{}% No number no space
\renewcommand*{\chaptermarkformat}{}% No number no space
\counterwithout{section}{chapter}
% This also results in \renewcommand*{\thesection}{\arabic{section}}
\counterwithout{figure}{chapter}
\counterwithout{table}{chapter}
\counterwithout{equation}{chapter}
\setcounter{secnumdepth}{\subparagraphnumdepth}% number every sectioning level from \part till \subparagraph (not recommended to do so)
\setcounter{tocdepth}{\subparagraphtocdepth}% add every sectioning level from \part till \subparagraph to the ToC (not recommended to do so)
\begin{document}
\tableofcontents
\chapter{Chapter about A}
\section{Thing a}
\subsection{Subthing aa}
\chapter{Chapter about B}
\section{Thing b}
\subsection{Subthing bb}
\chapter{Chapter about C}
\section{Thing c}
\subsection{Subthing cc}
\end{document}
If \counterwithout results in an error, you should either update your LaTeX or activate the commented \usepackage{chngcntr}.
Maybe you'd also want to change the spacing in the table of contents. But this would be another question.
