A little closer to production code by opening a "file" and checking for unexpected data:

use strict;
use warnings;
use Data::Dumper;

my $testData = <<DATA;
>James
>40
>James
>35
>James
>26
>James
>15
DATA

open my $fin, '<', \$testData or die "Can't open file: $!\n";

my %people;
my $lastPerson;

while (defined (my $line = <$fin>)) {
    chomp $line;

    if ($line =~ /^>(\d+)$/) {
        die "Unexpected data order: '$line'\n" if !defined $lastPerson
+;
        push @{$people{$lastPerson}}, $1;
        next;
    }

    if ($line =~ /^>(\w+)$/) {
        $lastPerson = $1;
        next;
    }

    warn "Don't know how to process '$line'\n";
}

print Dumper(\%people);
[download]

Prints:

$VAR1 = {
          'James' => [
                       '40',
                       '35',
                       '26',
                       '15'
                     ]
        };
[download]