\documentclass{article}
\usepackage{tabstackengine}
\stackMath
\newcommand\Frac[2]{\frac{\addstackgap{#1}}{\addstackgap{#2}}}
\begin{document}
\[
\setstacktabbedgap{1ex}
\bracketVectorstack{0\\0\\1\\1} -
\Frac{
\bracketMatrixstack{0&0&1&1}\bracketVectorstack{1\\1\\1\\1}
}{
\bracketMatrixstack{1&1&1&1}\bracketVectorstack{1\\1\\1\\1}
}
\bracketVectorstack{1\\1\\1\\1}
\]
\end{document}

The smaller size of \Frac
is also possible, with a small redefinition:
\documentclass{article}
\usepackage{tabstackengine}
\stackMath
\newcommand\Frac[2]{{%
\setstackgap{L}{.7\normalbaselineskip}\TABstackMathstyle{\scriptstyle}%
\frac{\addstackgap{#1}}{\addstackgap{#2}}%
}}
\begin{document}
\[
\setstacktabbedgap{1ex}
\bracketVectorstack{0\\0\\1\\1} -
\Frac{
\bracketMatrixstack{0&0&1&1}\bracketMatrixstack{1\\1\\1\\1}
}{
\bracketMatrixstack{1&1&1&1}\bracketMatrixstack{1\\1\\1\\1}
}
\bracketVectorstack{1\\1\\1\\1}
\]
\end{document}

$
causes TeX to leave math mode before your equation is finished. You should put it all the way at the end (after\end{bmatrix} }
). You may want to instead use\[...\]
or\begin{equation}...\end{equation}
instead of$...$
since this looks like a display equation rather than an inline equation.$
and got my desired outcome so my question no longer needs an answer