I fed your blob of code into perltidy, which complained about certain errors. I made minimal changes to satisfy perltidy, and wound up with the following:

%responce_array1                                    = ();
%responce_array2                                    = ();
$responce_array1{'REQUEST'}{'Items'}{0}{'ItemName'} = 'demoName0';
$responce_array1{'REQUEST'}{'Items'}{1}{'ItemName'} = 'demoName1';
$responce_array1{'REQUEST'}{'Items'}{2}{'ItemName'} = 'demoName2';
my $itemjson_ary1 = encode_json \%responce_array1;
$responce_array2{'item_json_array'} = $itemjson_ary1;
my $itemjson_ary2 = encode_json \%responce_array2;
print $itemjson_ary2;
$jsonary
  = '{"item_json_array":"{\"REQUEST\":{\"Items\":{\"1\":{\"ItemName\":
+\"demoName1\"},\"0\":{\"ItemName\":\"demoName0\"},\"2\":{\"ItemName\"
+:\"demoName2\"}}}}"}';
$dec_json_obj = decode_json $jsonary;
$temp         = $dec_json_obj->{item_json_array};
$dec_json_obj = decode_json $temp;
$temp         = $dec_json_obj->{REQUEST};

#hash form $dec_json_obj=&get_hash_to_str($temp);
$temp         = $dec_json_obj->{Items};
$dec_json_obj = &get_hash_to_str($temp);
$temp         = $dec_json_obj->{0};
$dec_json_obj = &get_hash_to_str($temp);
print $temp= $dec_json_obj->{ItemName};

sub get_hash_to_str($temp) {
    my $temp         = $_[0];
    my $enc_json_req = encode_json $temp;
    print " ";
    print $enc_json_req;
    print " ";
    my $dec_json_obj = decode_json $enc_json_req;
  return $dec_json_obj;
}

# Here i am sending values like these to retrieve
# 0-th element of item and get ItemName
#
$temp         = $dec_json_obj->{0};
$dec_json_obj = &get_hash_to_str($temp);
print $temp= $dec_json_obj->{ItemName};
[download]

I've made some minor spelling corrections.

• Declaring a new array automaticall makes it empty; declaring a new hash automatically makes it empty. You only need to initialize it to () if it previously had contents and you want to discard those old values.
• A variable with a % in front is a hash, not an array.
• You don't need to quote hash keys, unless they contain spaces or other characters not acceptable in a bareword hash key. My philosophy is to always use warnings and strict, and the computer will tell me when I have a dubious key value that needs quotes.
• If you have a hash with values {0], {1}, {2}, you should probably be using an array.
• Using ampersand (&) on subroutine calls used to be correct in Perl4 ... 1993 ... but is usually wrong in Perl5. If you have &, you should almost never have any arguments.
• Do you really want to have print $temp = ...; i.e., printing the return value of the assignment? The result is the same as $temp = ...; print $temp, but it's irregular, unusual, requires thinking about rather than being standard.
• In a subroutine, the normal way to set arguments is:my ( $var1, $var2 ) = @_;.
• The few print statements are odd ... looks like maybe they are an attempt at debugging?
• All variables are temporary; try using names relevant to the problem you are trying to solve, rather than relevant to the programming language solution you are implementing.
• I gave up on trying to figure out what that string thing in the middle was, with all the escaped double quotes. I mean, I can see you are trying to build up an encoded string representation of some object, but it is ugly, and difficult to verify as being right or wrong. I would suggest building up small pieces, and assembling them together. given the availability of single quote, double quote, qw{}, q{}, and qq{}, it shouldn't be necessary to escape quotes in strings.

my ( %response1, %response2 );
$response1{REQUEST}{Items}[0]{ItemName} = 'demoName0';
$response1{REQUEST}{Items}[1]{ItemName} = 'demoName1';
$response1{REQUEST}{Items}[2]{ItemName} = 'demoName2';

my $itemjson_ary1 = encode_json \%response1;
$response2{item_json_array} = $itemjson_ary1;

my $itemjson_ary2 = encode_json \%resposce2;
print $itemjson_ary2;

$jsonary
  = '{"item_json_array":"{\"REQUEST\":{\"Items\":{\"1\":{\"ItemName\":
+\"demoName1\"},\"0\":{\"ItemName\":\"demoName0\"},\"2\":{\"ItemName\"
+:\"demoName2\"}}}}"}';
$dec_json_obj = decode_json $jsonary;
$temp         = $dec_json_obj->{item_json_array};
$dec_json_obj = decode_json $temp;

$temp         = $dec_json_obj->{REQUEST}; # hash form 
$dec_json_obj = get_hash_to_str($temp);

$temp         = $dec_json_obj->{Items};
$dec_json_obj = get_hash_to_str($temp);

$temp         = $dec_json_obj->{0};
$dec_json_obj = get_hash_to_str($temp);

print $temp   = $dec_json_obj->{ItemName};

sub get_hash_to_str($temp) {
    my ( $stringify_me ) = @_;
    my $enc_json_req = encode_json $stringify_me;
    print " ";
    print $enc_json_req;
    print " ";
    my $dec_json_obj = decode_json $enc_json_req;
  return $dec_json_obj;
}

# Here i am sending values like these to retrieve
# 0-th element of item and get ItemName
#
$temp         = $dec_json_obj->[0];
$dec_json_obj = get_hash_to_str($temp);
print $temp   = $dec_json_obj->{ItemName};
[download]

Hope this helps a little.