but you're wrong in saying that __DATA__ can't be accessed twice

This is not what I said.  I said that you can only have one __DATA__ section (per package/file), not that you can only read it once.

The OP asked "how to use multiple sets of __DATA__", which I understood to mean something like this (also see the OP's sample code)

#!/usr/bin/perl

while (<DATA>) { print }

__DATA__
set 1
...
__DATA__
set 2
...
[download]

which obviously acts as only one DATA section:

$ ./887252.pl
set 1
...
__DATA__
set 2
...
[download]