You got your odd/even mixed up. The first element is at index 0, which is even. Also, enter splice and statement modifiers.

while( @array ) {
    my ( $even, $odd ) = splice @array, 0, 2;
    push @odd, $odd if defined $odd;
    push @even, $even if defined $even;
}
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It is worth noting that this would be buggy if the list could possibly have undefs in it (not the case here; but I'd put a comment to that effect in the code).

Obscure trickery for the fun of reducing it further, but don't do this at home:

( $even[ @even ], $odd[ @odd ] ) = splice @array, 0, 2 while @array;
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I shouldn't have used the names "@odd" and "@even", seeing as how we're supposed to be dealing with text. So here's a program which doesn't use those terribly misleading names, and deals with undefs.

#!/usr/bin/perl

use strict;
use warnings;

my @array = map{chr} ('32' .. '126');
push @array, undef, undef;
unshift @array, undef, undef, undef;

# split elements into two arrays
my (@arr0, @arr1);
while (@array > 1)
{
    my ($el0, $el1) = splice @array, 0, 2;
    push @arr0, $el0;
    push @arr1, $el1;
}

# get the last element if there is one
if(@array)
{
    push @arr0, shift @array;
}

# check that all elements were gotten
die "still elements left" if (@array);

print "arr0 = @arr0\narr1 = @arr1\n";

__END__
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If you do away with the conditionals like that, the temporaries in the while loop are superfluous.

my ( @arr0, @arr1 );

while( @array > 1 ) {
    push @arr0, shift @array;
    push @arr1, shift @array;
}
push @arr0, shift @array if @array;
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Or you could pull the conditional back into the loop:

my ( @arr0, @arr1 );

while( @array ) {
    push @arr0, shift @array;
    push @arr1, shift @array if @array;
}
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Makeshifts last the longest.