If you need to know that it ran then don't go for backticks... but reasonablekeith has correctly offered here the answer to the bit you said you needed - the exit code of the script - which you can control with the exit command.

#!/bin/sh
#
# shell_script.sh
#
MY_VAR="something"

./simulated_binary_executable.pl
echo "Returned $?"

echo "Exporting MY_VAR"
export MY_VAR

./simulated_binary_executable.pl
echo "Returned $?"

exit 0
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#!/usr/bin/perl
#
# simulated_binary_executable.pl
#

use strict;
use warnings;

if ( $ENV{MY_VAR} ) {
    print "MY_VAR is '$ENV{MY_VAR}'\n";
    exit 0;
}
else {
    print "MY_VAR is not set\n";
    exit 1;
}
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#!/usr/bin/perl
#
# Dummy of outer perl script that is calling shell script
#

use strict;
use warnings;

# You could use $? later instead of defining $return
my $return = system("./shell_script.sh");

if ($return == -1) {
    print "failed to execute: $!\n";
}
elsif ($return & 127) {
    printf "child died with signal %d, %s coredump\n",
        ($return & 127),  ($return & 128) ? 'with' : 'without';
}
else {
    printf "child exited with value %d\n", $return >> 8;
}
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#!/usr/bin/perl
#
# Dummy of outer perl script
# This time calling the binary without shell wrapper script
#

use strict;
use warnings;

#
# Export the environment variable for the child binary
# directly from Perl
#
$ENV{MY_VAR} = "something else";

my $binary = "./simulated_binary_executable.pl";

system($binary); # Run the external program

if ($? == -1) {
    print "Child '$binary' failed to execute: $!\n";
}
elsif ($? & 127) {
    printf "Child '%s' died with signal %d, %s coredump\n",
        $binary, ($? & 127),  ($? & 128) ? 'with' : 'without';
}
elsif  ($? == 0) {
    print "Child '%s' exited successfully.\n";
}
else {
    printf "Child '%s' exited with value %d\n", $binary, $? >> 8;
}
[download]

Short version: no, but do read the FAQ.