Another way to do this is to treat your data as an "in memory" file (see perldoc -f open). Setting the INPUT_RECORD_SEPARATOR variable to a reference to an integer will make the readline operator read that many bytes, or as many as it can until EOF. Putting that all together gives us:

#!/usr/bin/perl

use warnings;
use strict;

my $string = "x" x 100_400;

{ 
  open ( my $sth, "<", \$string ) or die $!;
  local $/ = \1_000;
  while ( my $chunk = <$sth> ) {
    print $chunk, "\n";
  }
}
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