Here's a somewhat simpler and more idiomatic version of your code that does the same thing that may help you. Ask about anything you do not understand:

#!/usr/bin/perl -w
use strict;

my( @arrayx, @arrayy, @arrayz );

while (<>) {
# Find x, y, z coordinates and store in separate arrays

    if ($_ =~ /^ATOM/) {
        my @line = $_ =~ m/^(.....).(.....).(....).(...)..(....)....(.
+.......)(........)(........)/;

        ## using push mean you don't have to count because ...
        push @arrayx, $line[5];
        push @arrayy, $line[6];
        push @arrayz, $line[7];
    }

}
close *ARGV; ## prevent confusing error message suffixes

# Calculate distance between all atom coordinates
## ... $#xxx gives you the highest index in array @xxx

foreach my $i ( 0 .. $#arrayx ) { 

    foreach my $j ( $i + 1 .. $#arrayx ) {

        my $dist = sqrt(
                ($arrayx[$i] - $arrayx[$j])**2 +
                ($arrayy[$i] - $arrayy[$j])**2 +
                ($arrayz[$i] - $arrayz[$j])**2
            );

        ## Adding $i and $j to your output will let you know what that
+ output is.
        print "$i <> $j : $dist\n";

    }
}
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