Not directly related to your issue, but some comments anyway, in no particular order:

while (<FILE1>)
        {    $hline=<FILE1>;
                push (@hlines,$hline);
        }
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is the same as

@hlines = <FILE1>;
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Also,

my $FILE1;
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probably doesn't do what you think. $FILE1 is not the same as the FILE1 you use in your open statement. Either change all places where you use a bare FILE1 to $FILE1 (which would make it a lexical filehandle), or just remove the my $FILE1; declaration.

In your open call, you don't have to put quotes around the filename. It is however a good idea to use the three-argument form of open, in case someone tries to use something like >datafile as filename and causes you to overwrite the data file that way. So instead of

open (FILE1, "$f1") or die "Argh! File not opened! $!";
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use something like

open (FILE1, "<", $f1) or die "Argh! File not opened! $!";
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In your read subroutine, you're using the variable $a without declaring it. With almost any other name this would have caused a compile time error, but $a and $b are special. For that reason, it's best to avoid using either of those names.

$xh, $yh, $zh, $x1, $y1, $z1, $i, $r and $r2 are only used within your dist subroutine, so you should only declare them in there, and not at the global level

read is also the name of a built-in function, so it's usually better to avoid using it for any of your own functions unless you explicitely want to override it

Your shebang line is showing /root/bin/perl as the path to perl. Since /root is usually only accessible by root, this means that you're developing and testing this code as root, which is generally not recommended.

The use subs; is useless here as far as I can tell.

Since the two file reads are really unrelated, you can reuse the same filehandle for both of them

Making the changes above, the code looks something like this:

#!/root/bin/perl
use strict;
use warnings;

sub square;
sub read_filename;
sub dist;

my $FILE;

my $f1 = read_filename;
open ($FILE, "<", $f1) or die "Argh! File not opened! $!";
my @hlines = <$FILE>;
close $FILE;

my $f2 = read_filename;
open ($FILE, "<", $f2) or die "Argh! Second file not opened! $!";
my @plines = <$FILE>;
close $FILE;

print @hlines;
print @plines;
    
dist;
exit;

sub dist
{
    my $i=0;
    my ($r, $r2);

    foreach my $hline (@hlines)
    {
        my $xh = substr $hline,31,8;
        my $yh = substr $hline,40,8;
        my $zh = substr $hline,48,8;
        foreach my $pline (@plines)
        {
            print '"',$pline,'": len=',length($pline),"\n";
            my $x1 = substr $pline,31,8;
            my $y1 = substr $pline,40,8;
            my $z1 = substr $pline,48,8;
                                    
            $i++;
            $r2=((square($xh-$x1))+(square($yh-$y1))+(square($zh-$z1))
+);
            $r=sqrt($r2);
            print "$i. $r2 and square root is $r\n";
            if(($r<=2.4)&&($r>=3.6))
                { print "match!"}
        }
    }
    print $i;
    return 1;
}

sub square
{
my $num = shift;
my $sq = $num * $num;
return $sq;
}

sub read_filename
{
print "Enter file name: ";
my $fn = <STDIN>;
chomp $fn;
return $fn;
}
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This code can stil be further improved, but it should be a nice step in the right direction.