You have warnings on! Why don't you read the warning you get? Useless use of exponentiation (**) in void context means that your $j**3 line isn't doing anything with its return value. Perhaps you meant $j **= 3 which would set $j to itself cubed.

I assume you have a sum() function defined somewhere. And your variable scoping leaves something to be desired -- specifically, proper variable scoping. Rewriting only the scoping of your program, I'd do:

my @numbers;

for my $i (100 .. 999) {
  my @num = split(//, $i);
  foreach my $j (@num) {
    $j **= 3;
  }
  if (sum(@num) == $i) {
    push @numbers, $i;
  }
}
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I have another comment. Why did you use 'for' in one loop and 'foreach' in another? They're not different things, and you're not using the C-style loop, so there's no reason one of them should have a different name from the other.

All in all, I would rewrite this program to use far fewer temporary variables:

my @numbers = grep { sum(map { $_ ** 3 } split //) == $_ } 100 .. 999;
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That might be a bit too succinct for you, but it does the same thing as yours.