local backups the variable being localized and creates a new one. The old one still exists anonymously. That means the original STDOUT still has file descriptor one open. open uses the first available file descriptor — probably three here — and associates it with STDOUT.

#!/usr/bin/perl -l

my $file = "file.txt";

print "before";
{
   print STDERR "fileno before local: ", fileno(STDOUT);

   local *STDOUT;
   open (STDOUT, '>', $file) or die;

   print STDERR "fileno after open: ", fileno(STDOUT);

   system("ls");
}

print "after";
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before
fileno before local: 1
fileno after open: 3
740300.pl
after
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ls takes fd0 as its STDIN, fd1 as its STDOUT and fd2 as its STDERR, which are inherited from the Perl script and unchanged by the local *STDOUT;. You need to change where fd1 points, not where STDOUT points.

#!/usr/bin/perl -l

my $file = "file.txt";

print "before";
{
   print STDERR "fileno before local: ", fileno(STDOUT);

   open (my $old_stdout, '>&', *STDOUT) or die;
   close (STDOUT);
   open (STDOUT, '>', $file) or die;

   print STDERR "fileno after open: ", fileno(STDOUT);

   system("ls");

   close (STDOUT);
   open (STDOUT, '>&', $old_stdout) or die;
}

print "after";
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$ perl 740300.pl
before
fileno before local: 1
fileno after open: 1
after

$ cat file.txt
740300.pl
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It might be easier just to use open2 or open3. Just pass them the handles and they'll do the duping (>&) for you.

#!/usr/bin/perl -l

use IPC::Open3 qw( open3 );

my $file = "file.txt";

print "before";

{
   open (my $fh, '>', $file) or die;
   my $pid = open3('<&STDIN', '>&'.fileno($fh), '>&STDERR', 'ls');
   waitpid($pid, 0);
}

print "after";
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Update: Fixed some errors in the fd numbers.