I'm thinking that line 5 is print $1   "\n";. You skipped a comma there, so Perl interprets that as printing to a filehandle in $1. You likely mean:

open (FILE,"abc345.txt") or die $!;
       while (<FILE>){
       $_ = /((^Query = .*)|(^\>.*)|(Length = .*))/i;
       print "$1\n";
       print "$2\n";
       print "$3\n";
       print "$4\n";
      
   }
  close (FILE) ;
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or

open (FILE,"abc345.txt") or die $!;
       while (<FILE>){
       $_ = /((^Query = .*)|(^\>.*)|(Length = .*))/i;
       print $1, "\n";
       print $2, "\n";
       print $3, "\n";
       print $4, "\n";
      
   }
  close (FILE) ;
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See print for info on proper syntax.