A Full Stop Suppresses the Delimiter Between Citations

It's actually simpler: the period in “ff.” should not be sentence ending, so the rules of LaTeX say it should be followed by \@.

\begin{filecontents*}{\jobname.bib}
@book{Sievers1893,
author = "Eduard Sievers",
title = "Altgermanische Metrik",
date = "1893",
publisher = "Max Niemeyer",
address = "Halle",
}

@book{Kloekhorst2008,
author = "Alwin Kloekhorst",
title = "Etymological Dictionary of the Hittite Inherited Lexicon",
date = "2008",
publisher = "Brill",
address = "Leiden",
}
\end{filecontents*}

\documentclass[a4paper,12pt]{article}
\usepackage[authordate,backend=biber]{biblatex-chicago}
\addbibresource{\jobname.bib}

\begin{document}

Here be dragons \autocites[57ff.\@]{Sievers1893}[33]{Kloekhorst2008}.

\printbibliography
\end{document}

(I used filecontents just to make the example self-contained, as usual.)

You should better use \psqq instead of an hard coded ff and customize it to your needs. But at least you should use \adddot instead of a period:

\documentclass[a4paper,12pt]{article}
\usepackage[english]{babel}
\usepackage[authordate,backend=biber]{biblatex-chicago}
\addbibresource{test.bib}

\DefineBibliographyStrings{english}{%
  sequens = {f\adddot},
  sequentes = {ff\adddot},
}

\renewcommand\sqspace{}

\begin{document}

Here be dragons \autocites[57\psqq ]{Sievers1893}[33]{Kloekhorst2008}.

\autocites[57ff\adddot ]{Sievers1893}[33]{Kloekhorst2008}.

\printbibliography
\end{document}