Your markers look like they are fixed strings, so

    if( $_ eq "--end foo--\n"){
    # . . .
    }
[download]

seems like a better test. Leave off "\n" if you chomp it.

A couple more approaches come to mind. If the file is not too big, you can abuse $/ to get the content in two gulps.

my $interesting_bit;
{
    # maybe open here . . .
    local $/ = '--start foo--' . "\n";
    $interesting_bit = <FILE>;
    local $/ = '--end foo--' . "\n";
    $interesting_bit = <FILE>;
    # . . . and close here
}
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That could burden memory.

A third way is similar to yours, but uses the flip-flop operator to condense the code.

my $interesting_bit;
{
    # maybe open here . . .
    local $_;
    while (<FILE>) {
        if ($_ eq "--start foo--\n" .. $_ eq "--end foo--\n"  ) {
            $interesting_bit .= $_;
        }
        elsif ( $interesting_bit ) { last }
    }
    # . . . and close
}
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This makes the same assumption about there being only one interesting bit. If the variable is populated while the flip-flop is false, we've run past the end marker and can quit reading.

I just noticed that I've changed the $interesting_bit variable from a flag in your code to an accumulator for the content. If you just set it true in the lhs of the flip-flop and do your processing in place of my .= operation, all will be well.