Alternative to \substack{} for a related purpose

Just use tabular for the text, or replace tabular with array for mathematics. You could also consider the Bmatrix environment from the amsmath package for the latter case.

\documentclass{article}
\begin{document}
\[
\left\{ 
\begin{tabular}{c}
TeX.SX users that \\
ask questions on \\
this site
\end{tabular}
\right\}
\longleftrightarrow
\left\{ 
\begin{tabular}{c}
TeX users in the \\
world using \\
TeX.SX
\end{tabular}
\right\}
\]
\end{document}

here are two possibilities, using amsmath. the spacing inside the braces is different, which may influence your preference.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[
 \left\{
  \begin{array}{c}
   a + b\\
   c + d\\
   e + f
  \end{array}
 \right\}
\quad\longleftrightarrow\quad
 \left\{
  \begin{array}{c}
   g + h\\
   i + j\\
   k + l
  \end{array}
 \right\}
\]

\[
 \begin{Bmatrix}
   a + b\\
   c + d\\
   e + f
 \end{Bmatrix}
\quad\longleftrightarrow\quad
 \begin{Bmatrix}
   g + h\\
   i + j\\
   k + l
 \end{Bmatrix}
\]
\end{document}

and the output:

For a somewhat simpler solution you could use something like

\begin{align*}
\begin{Bmatrix}
& \text{hi}  \\
& \text{hello}
\end{Bmatrix}
\longleftrightarrow
\begin{Bmatrix}
& \text{The droids} \\
& \text{lorrem lipsum}
\end{Bmatrix}
\end{align*}

to get the exact arrow style you would have to use tikz and the arrows libary.