Can \ExplSyntaxOn and \ExplSyntaxOff be nested?

No, you can't. There's no stack implemented.

When \ExplSyntaxOn is encountered, some characters have their category code are changed, but their previous category code is stored in the definition of \ExplSyntaxOff (that does nothing, by default).

When \ExplSyntaxOff is next encountered, it resets the category codes to the previous states and redefines itself to do nothing.

Here's an example:

\documentclass{article}
\usepackage{expl3}

\ExplSyntaxOn
\tl_show:N \ExplSyntaxOff

\ExplSyntaxOn
\tl_show:N \ExplSyntaxOff

\ExplSyntaxOff
\tl_show:N \ExplSyntaxOff

\ExplSyntaxOff

\stop

And here's the output on the terminal: you see that the expl3 codes are not valid any longer after the first \ExplSyntaxOff and \tl_show:N raises an error.

> \ExplSyntaxOff=\char_set_catcode:nn {9}{10}\char_set_catcode:nn
{32}{10}\char_set_catcode:nn {34}{12}\char_set_catcode:nn
{38}{4}\char_set_catcode:nn {58}{12}\char_set_catcode:nn
{94}{7}\char_set_catcode:nn {95}{8}\char_set_catcode:nn
{124}{12}\char_set_catcode:nn {126}{13}\tex_endlinechar:D =13\scan_stop:
\bool_set_false:N \l__kernel_expl_bool \cs_set_protected:Npn \ExplSyntaxOff
{}.
<recently read> }

l.5 \tl_show:N \ExplSyntaxOff

? 
> \ExplSyntaxOff=\char_set_catcode:nn {9}{10}\char_set_catcode:nn
{32}{10}\char_set_catcode:nn {34}{12}\char_set_catcode:nn
{38}{4}\char_set_catcode:nn {58}{12}\char_set_catcode:nn
{94}{7}\char_set_catcode:nn {95}{8}\char_set_catcode:nn
{124}{12}\char_set_catcode:nn {126}{13}\tex_endlinechar:D =13\scan_stop:
\bool_set_false:N \l__kernel_expl_bool \cs_set_protected:Npn \ExplSyntaxOff
{}.
<recently read> }

l.8 \tl_show:N \ExplSyntaxOff

? 
! Undefined control sequence.
l.11 \tl
        _show:N \ExplSyntaxOff
? 

\ExplSyntaxOn and \ExplSyntaxOff are switches that change certain category codes to allow for a different coding syntax. They're synonymous to the \makeatletter...\makeatother pairs used to allow @'s in macros.

A showcase of something similar using font switches may be conclusive:

\documentclass{article}

\begin{document}

\bfseries
First % ... lots of stuff 1 ...
\bfseries
Second % ... lots of stuff 2 ...
\mdseries
Third % ... lots of stuff 3 ...
\mdseries
Last % ... lots of stuff last ...

\end{document}

One could consider \mdseries reverting what \bfseries has done, but that doesn't imply they are nestable.

Switches are not the same as grouping, which provides a scope as well as nesting.