The statements
    my %index2;
    eval{ %index2 = ( $index ) };
are equivalent to the statement
    my %index2 = ( $index );
with the only difference being the time at which they are evaluated: the eval statement is evaluated at run time; the other statement is evaluated at compile time | the eval statement is executed such that run-time errors are captured; both statements must be syntactically correct at compile time. (Thanks to Anonymonk for pointing out below the original confusion of this statement. As always, see The Docs for full details!)

But consider
    my %index2 = ( $index );
This statement attempts to initialize a hash with the single string  $index used as a key (and with no associated value) and not with a key/value pair. That's not going to work. Instead, consider

c:\@Work\Perl\monks>perl -wMstrict -MData::Dumper -le
"my $index = q{1 => 'foo',  2 => 'bar',  3 => 'baz'};
 my %hash = ( $index => 'foobar' );
 print Dumper \%hash;
"
$VAR1 = {
          '1 => \'foo\',  2 => \'bar\',  3 => \'baz\'' => 'foobar'
        };
[download]

Doing a full string eval would have given you the result you expected:

c:\@Work\Perl\monks>perl -wMstrict -MData::Dumper -le
"my $index = q{1 => 'foo',  2 => 'bar',  3 => 'baz'};
 my %hash;
 eval qq{ %hash = ($index); };
 print Dumper \%hash;
"
$VAR1 = {
          '1' => 'foo',
          '3' => 'baz',
          '2' => 'bar'
        };
[download]

c:\@Work\Perl\monks>perl -wMstrict -MData::Dumper -le
"my $index = q{1 => 'foo',  2 => 'bar',  3 => 'baz'};
 my %hash = eval qq{ ($index) };
 print Dumper \%hash;
"
$VAR1 = {
          '1' => 'foo',
          '3' => 'baz',
          '2' => 'bar'
        };
[download]

See eval for a discussion of the differences between the
    eval { statement; ...; };
and
    eval "statement; ...;";
forms (referred to respectively as  eval BLOCK and  eval EXPR in the doc).

the eval statement is evaluated at run time; the other statement is evaluated at compile time.

I don't think that's accurate for the eval BLOCK form... perhaps you're thinking of evaling a string? Also it depends on what you mean by "evaluated" (compiled vs. executed).

Try %index = eval "$index"

( actually the double quotes are even not necessary here )

use strict;
use warnings;
use Data::Dumper;
my $index = q{
      1 => 'foo',
      2 => 'bar',
      3 => 'baz',
   };
my %index3 = eval $index;
print Dumper \%index3;
[download]

$VAR1 = {
          '1' => 'foo',
          '3' => 'baz',
          '2' => 'bar'
        };
[download]