First of all, that's a bad design. setspace is designed to avoid the problem of setting \baselinestretch to the whole document, so that one can maintain footnotes and block quotes in single spacing.
You can use footmisc package and change \footnotelayout
\documentclass[12pt]{article}
\begin{filecontents}{\jobname.bib}
@book{test1,
author = {John Doe},
title = {Example book to be cited},
address = {Nowhere},
publisher = {Some Publisher},
year = {1990},
}
\end{filecontents}
\usepackage{biblatex-chicago}
\addbibresource{\jobname.bib}
\usepackage{setspace}
\usepackage{footmisc}
\renewcommand{\footnotelayout}{\doublespacing}
\newcommand\lipsum{Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Praesent egestas lectus eu nisi placerat sodales. Praesent a vestibulum tortor.
Aliquam sem lectus, dapibus in aliquam sit amet, euismod sed orci.
Phasellus posuere placerat mi ut accumsan. Proin eu interdum nunc.
Class aptent taciti sociosqu ad litora torquent per conubia nostra,
per inceptos himenaeos. Aenean semper elementum nulla, ac mattis erat
commodo ac. Etiam mi tellus, lobortis vitae ultrices quis, tempus at ipsum.
Donec diam leo, egestas at vestibulum eget, vestibulum eu est. Cras purus
augue, aliquet eget gravida id, aliquet quis lacus. Nullam accumsan,
metus sed placerat tincidunt, lacus leo pulvinar ipsum, vitae suscipit
mi mi auctor odio. Donec tristique, nisi ut rhoncus molestie, est
justo ultricies nulla, et consequat nunc est quis sem.}
\begin{document}
\singlespacing
\lipsum
\lipsum\autocite[\lipsum][100]{test1}
\lipsum
\end{document}