Well, at least writing these was enjoyable. Here are two solutions, both slightly adjusted to have data that does not evenly fit the number of bins.

modulo.pl:

#!/usr/bin/perl

use strict;
use warnings;

my @array = 1 .. 14;

use constant BINS => 4;
my @bins = ();

for my $i (0 .. $#array) {
  print "i = $i:\t$array[$i]\n";
  push @{$bins[$i % BINS]}, $array[$i];
}

use Data::Dumper;
print Dumper \@bins;
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sample output:

i = 0:  1
i = 1:  2
i = 2:  3
i = 3:  4
i = 4:  5
i = 5:  6
i = 6:  7
i = 7:  8
i = 8:  9
i = 9:  10
i = 10: 11
i = 11: 12
i = 12: 13
i = 13: 14
$VAR1 = [
          [
            1,
            5,
            9,
            13
          ],
          [
            2,
            6,
            10,
            14
          ],
          [
            3,
            7,
            11
          ],
          [
            4,
            8,
            12
          ]
        ];
[download]

slice.pl

#!/usr/bin/perl

use strict;
use warnings;

my @array = 1 .. 14;

use constant BINS => 4;
my @bins = ();

for my $i (0 .. (BINS - 1)) {
  push @bins, [@array[grep {defined $array[$_]}
              map {BINS * $_ + $i} 0 .. (@array / BINS)]];
}

use Data::Dumper;
print Dumper \@bins;
[download]

sample output:

$VAR1 = [
          [
            1,
            5,
            9,
            13
          ],
          [
            2,
            6,
            10,
            14
          ],
          [
            3,
            7,
            11
          ],
          [
            4,
            8,
            12
          ]
        ];
[download]