apacite implements the citation and bibliography style from the 6th edition of the APA publication manual.
APA style is an author-year style and does not involve numbering entries in the bibliography as far as I know. Indeed numbered bibliographies will look out of place with an author-year citation style.
Now that you have been warned sufficiently that this is not APA compliant and a bad idea in general, let's attempt to get a numbered bibliography.
With natbibapa apacite delegates control over the bibliography environment (thebibliography) to natbib, so we have to fiddle with natbib's internal commands to make it produce a numbered bibliography.
There is one problem here: apacite's bibliography style does not calculate the longest numeric label that we need to format the list properly. Because I'm lazy I just hard-coded a value that seemed sensible below (1em, but you may need a higher value if the numbers get longer).
\documentclass[british]{article}
\usepackage[T1]{fontenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[natbibapa]{apacite}
\makeatletter
\renewcommand\@biblabel[1]{[#1]}%
\renewcommand\NAT@bibsetnum[1]{%
\setlength{\labelwidth}{1em}%
\setlength{\labelsep}{2\labelsep}%
\setlength{\leftmargin}{\labelwidth}
\addtolength{\leftmargin}{\labelsep}%
\setlength{\itemsep}{\bibsep}\setlength{\parsep}{\z@}%
\ifNAT@openbib
\addtolength{\leftmargin}{\bibindent}%
\setlength{\itemindent}{-\bibindent}%
\setlength{\listparindent}{\itemindent}%
\setlength{\parsep}{0pt}%
\fi
}
\newcommand*{\numbibliography}{%
\let\@bibsetup\NAT@bibsetnum
\let\@biblabel\NAT@biblabelnum
\bibliography}
\makeatother
\begin{document}
\citet{apa6:ch7-ex38}
\bibliographystyle{apacite}
\numbibliography{apa5ex}
\end{document}

It should be noted that the current edition of the APA publication manual is the 7th edition. The only 7th-edition APA style implementation I know of is biblatex-apa.
With biblatex-apa you could use the following definitions to get a numbered bibliography (see Using bibstyle=numeric but displaying autho, year without brackets in the bibliography)
\documentclass[british]{article}
\usepackage[T1]{fontenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=apa, backend=biber]{biblatex}
\addbibresource{biblatex-examples.bib}
\defbibenvironment{bibliography}
{\setlength{\leftmargin}{\bibhang}%
\setlength{\itemindent}{-\leftmargin}%
\setlength{\labelsep}{\biblabelsep}%
\setlength{\itemsep}{\bibitemsep}%
\setlength{\parsep}{\bibparsep}%
\begin{enumerate}}
{\end{enumerate}}
{\item}
\begin{document}
\autocite{sigfridsson}
\printbibliography
\end{document}
Note that with this setting there is no way to access these label numbers later. If you wanted that, you'd need to activate the option labelnumber and
\DeclareFieldFormat{labelnumberwidth}{\mkbibbrackets{#1}}
\DeclareFieldFormat{shorthandwidth}{\mkbibbrackets{#1}}
\defbibenvironment{bibliography}
{\list
{\printtext[labelnumberwidth]{%
\printfield{labelprefix}%
\printfield{labelnumber}}}
{\setlength{\labelwidth}{\labelnumberwidth}%
\setlength{\leftmargin}{\labelwidth}%
\setlength{\labelsep}{\biblabelsep}%
\addtolength{\leftmargin}{\labelsep}%
\setlength{\itemsep}{\bibitemsep}%
\setlength{\parsep}{\bibparsep}}%
\renewcommand*{\makelabel}[1]{\hss##1}}
{\endlist}
{\item}
apaciteimplements APA style as described in the 6th edition of the APA publication manual. As far as I know the manual does not mention numbered references, so it is no surprise thatapacitedoesn't support numbered references out of the box. The problem with combining numbered references with author-year citations in the text is that the numbers in the bibliography are going to be completely unmotivated and detached from anything else that is going on in your document. ... – moewe Apr 11 '20 at 12:15