I provide two environments, mybraceleft and mybraceright, for conveniently bracing horizontal material. They both produce horizontal material (you can use them in the middle of a paragraph) of just the required size.
One notable property of my solution as compared to David's one is that it produces horizontal material that tightly wraps the tabular environments, or whatever you decide to use in their place (you don't have to guess the 0.45\textwidth length). If you reduce the width of these elements, you'll see that my braces remain at the customizable distance \myhsep from the braced contents, which is not the case with David's solution (you'll have a larger gap between braces and contents because the minipage will have more whitespace inside, on the left and on the right of the tabular material).
A tricky part of the work with the requirements I set is to find the appropriate width for centering two tabular environments and two captions that go together. For this, I use the eqparbox package. This package relies on proper use of tags (for each tag, it finds the maximum width and allows you to use it as a normal length—this is extremely convenient, great package!). Watch for the tags in the source, they go by three (one for a minipage that tightly encloses two tabular environments and their associated captions). This process requires to compile the document at least twice.
With no table environment
In this solution, the table environment isn't used, therefore nothing floats. This is mainly boxing work. \captionof from the caption package is used to produce the customary appearance of standard table captions.
\documentclass{article}
\usepackage[hscale=0.65]{geometry} % enlarge margins a little bit for the example
\usepackage{eqparbox}
\usepackage{caption}
\makeatletter
\newsavebox{\mybox}
\newlength{\myhsep} % Horizontal separation between brace and contents
\setlength{\myhsep}{0.7em}
\newenvironment{mybraceleft}{%
\begin{lrbox}{\mybox}
}{%
\end{lrbox}%
$\m@th \left\{ \kern\myhsep \vcenter{\hbox{\usebox{\mybox}}} \right.$%
\kern-\nulldelimiterspace
\ignorespacesafterend
}
\newenvironment{mybraceright}{%
\begin{lrbox}{\mybox}
}{%
\end{lrbox}%
\leavevmode
\kern-\nulldelimiterspace
$\m@th \left. \vcenter{\hbox{\usebox{\mybox}}} \kern\myhsep \right\}$%
\ignorespacesafterend
}
\makeatother
\begin{document}
\noindent
\begin{mybraceleft}
\begin{minipage}{\eqboxwidth{tag1}}
\centering
\begin{eqminipage}[c]{tag1}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\captionof{table}{Table \#1}
\bigskip
\begin{eqminipage}[c]{tag1}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\captionof{table}{Table \#2}
\end{minipage}
\end{mybraceleft}% spaces would be ignored here anyway because of the
% \ignorespacesafterend we used
\hfill
\begin{mybraceleft}
\begin{minipage}{\eqboxwidth{tag2}}
\centering
\begin{eqminipage}[c]{tag2}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\captionof{table}{Table \#3}
\bigskip
\begin{eqminipage}[c]{tag2}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\captionof{table}{Table \#4}
\end{minipage}
\end{mybraceleft}
\vspace{2cm}
\noindent
\begin{mybraceleft}
\begin{minipage}{\eqboxwidth{tag3}}
\centering
\begin{eqminipage}[c]{tag3}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\captionof{table}{Table \#5}
\bigskip
\begin{eqminipage}[c]{tag3}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\captionof{table}{Table \#6}
\end{minipage}
\end{mybraceleft}% spaces would be ignored here anyway because of the
% \ignorespacesafterend we used
\hfill
\begin{mybraceright}
\begin{minipage}{\eqboxwidth{tag4}}
\centering
\begin{eqminipage}[c]{tag4}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\captionof{table}{Table \#7}
\bigskip
\begin{eqminipage}[c]{tag4}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\captionof{table}{Table \#8}
\end{minipage}
\end{mybraceright}
\end{document}

Wrapping braced material inside tables
This is close to the previous example, the only difference is that we allow two big chunks (each containing four tabular environments) to float. Thus, we use the table environment and can use \caption instead of \captionof. No need for the caption package in this case. This is similar to what David did, except for the input syntax (mybraceleft and mybraceright here) and horizontal spacing around the braces: my braces stay at a constant, customizable distance from the braced material (\myhsep).
As already mentioned, this requires two compilation runs to allow eqparbox to do its work.
\documentclass{article}
\usepackage[hscale=0.65]{geometry} % enlarge margins a little bit for the example
\usepackage{eqparbox}
\makeatletter
\newsavebox{\mybox}
\newlength{\myhsep} % Horizontal separation between brace and contents
\setlength{\myhsep}{0.7em}
\newenvironment{mybraceleft}{%
\begin{lrbox}{\mybox}
}{%
\end{lrbox}%
$\m@th \left\{ \kern\myhsep \vcenter{\hbox{\usebox{\mybox}}} \right.$%
\kern-\nulldelimiterspace
\ignorespacesafterend
}
\newenvironment{mybraceright}{%
\begin{lrbox}{\mybox}
}{%
\end{lrbox}%
\leavevmode
\kern-\nulldelimiterspace
$\m@th \left. \vcenter{\hbox{\usebox{\mybox}}} \kern\myhsep \right\}$%
\ignorespacesafterend
}
\makeatother
\begin{document}
\begin{table}
\centering
\begin{mybraceleft}
\begin{minipage}{\eqboxwidth{tag1}}
\centering
\begin{eqminipage}[c]{tag1}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\caption{Table \#1}
\bigskip
\begin{eqminipage}[c]{tag1}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\caption{Table \#2}
\end{minipage}
\end{mybraceleft}% spaces would be ignored here anyway because of the
% \ignorespacesafterend we used
\hfill
\begin{mybraceleft}
\begin{minipage}{\eqboxwidth{tag2}}
\centering
\begin{eqminipage}[c]{tag2}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\caption{Table \#3}
\bigskip
\begin{eqminipage}[c]{tag2}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\caption{Table \#4}
\end{minipage}
\end{mybraceleft}
\end{table}
\begin{table}
\centering
\begin{mybraceleft}
\begin{minipage}{\eqboxwidth{tag3}}
\centering
\begin{eqminipage}[c]{tag3}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\caption{Table \#5}
\bigskip
\begin{eqminipage}[c]{tag3}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\caption{Table \#6}
\end{minipage}
\end{mybraceleft}% spaces would be ignored here anyway because of the
% \ignorespacesafterend we used
\hfill
\begin{mybraceright}
\begin{minipage}{\eqboxwidth{tag4}}
\centering
\begin{eqminipage}[c]{tag4}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\caption{Table \#7}
\bigskip
\begin{eqminipage}[c]{tag4}
\begin{tabular}{@{}|c|c|c|c|c|c|c|c|@{}}
\hline
a & b & c & d & e & f & g & h\\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
a & b & c & d & e & f & g & h\\
\hline
\end{tabular}
\end{eqminipage}%
\caption{Table \#8}
\end{minipage}
\end{mybraceright}
\end{table}
\end{document}
