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

my @stack = ();
my %ops = ( '+' => \&sum, '-' => \&difference,
            '*' => \&product, '/' => \&quotient,
            'p' => sub { print $_, "\n" foreach @stack; },
            'q' => sub { exit 0; }
          );

while(<>) {
  chomp;
  if (exists $ops{$_}) {
    $ops{$_}->();
  # The badass regex is out of the Perl Cookbook, page 44
  # Makes sure non-ops are valid numbers
  } elsif ($_ =~ /^([+-]?)(?=\d|\.\d)\d*(\.\d*)?([Ee]([+-]?\d+))?$/) {
    push @stack, $_;
  } else {
    warn "Bad input: $_\n";
  }
}

sub pop_stack {
  my $rhs = pop @stack;
  my $lhs = pop @stack;
  if ((! defined $lhs) || (! defined $rhs)) {
    warn "Not enough values on stack!\n";
    push @stack, $rhs if defined $rhs;
    return undef;
  }
  return $lhs, $rhs;
}

sub sum {
  my ($lhs, $rhs) = pop_stack;
  return unless defined $lhs;
  my $res = $lhs + $rhs;
  print "$res \n";
  push @stack, $res;
}

sub difference {
  my ($lhs, $rhs) = pop_stack;
  return unless defined $lhs;
  my $res = $lhs - $rhs;
  print "$res \n";
  push @stack, $res;
}

sub product {
  my ($lhs, $rhs) = pop_stack;
  return unless defined $lhs;
  my $res = $lhs * $rhs;
  print "$res \n";
  push @stack, $res;
}

sub quotient {
  my ($lhs, $rhs) = pop_stack;
  return unless defined $lhs;
  my $res = $lhs / $rhs;
  print "$res \n";
  push @stack, $res;
}
[download]

Well, here's a little bit of golf right off the bat. Since we are checking the input, we can get away with an eval without fear:

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

my @stack = ();
my %ops = ( '+' => \&calc, '-' => \&calc, 
            '*' => \&calc, '/' => \&calc, 
            'p' => sub { print $_, "\n" foreach @stack; },
            'q' => sub { exit 0; }
          );      

while(<>) {
  chomp;  
  if (exists $ops{$_}) {
    $ops{$_}->($_);
  # The badass regex is out of the Perl Cookbook, page 44
  } elsif ($_ =~ /^([+-]?)(?=\d|\.\d)\d*(\.\d*)?([Ee]([+-]?\d+))?$/) {
    push @stack, $_;
  } else {
    warn "Bad input: $_\n";
  }
}

sub calc {
  my $op = shift;
  my $rhs = pop @stack; 
  my $lhs = pop @stack; 
  if ((! defined $lhs) || (! defined $rhs)) {
    warn "Not enough values on stack!\n";
    push @stack, $rhs if defined $rhs;
    return undef;
  }
  my $res = eval $lhs.$op.$rhs;
  print "$res\n";
  push @stack, $res;
}
[download]

Fix: I had the rhs and lhs backwards in the eval originally

That is far above me but I will get there.

-Variable_B

That's above me but I am getting there!

-Variable_B

You have to be cautious of using eval(). It's quite possible for some joker to come along and type "rm / -rf" and run your program. That said, it's also the easiest way to get the code to work. Some others have suggested some very good checks to take care of malicious use of eval().