Yes, provided everything in your program (and its modules) use die instead of printing to STDERR. (Well, even then there are ways, but I won't go into them now).

What you need to do is use a die handler. Here's a complete program that logs errors to a file:

#!/opt/perl/bin/perl

use warnings 'all';
use strict;

{   my ($fh, $logfile);

    INIT {
        $logfile = "/var/log/whatever/errors";
        open $fh => ">> $logfile" or die "Failed to open $logfile: $!\
+n";
        $SIG {__DIE__} = sub {
            print $fh @_;
            print $fh "\n" unless $_ [-1] =~ /\n\z/;
        };
    }

    END {
        $fh and (close $fh or warn "Failed to close $logfile: $!\n");
    }
}

print "Hello, world\n";
die "Meep, meep!";
print "I should be dead....\n";
__END__
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-- Abigail