You are using a relative directory path, and your current directory is not what you think. Use the full path to your "temp" directory, or better, use the File::Temp module to create a filename for a temporary file:

#!/usr/bin/perl -wT
use strict;
BEGIN {
    $|=1;
    print "Content-type: text/html\n\n";
    use CGI::Carp('fatalsToBrowser');
}
use CGI;
use LWP::Simple;

use File::Temp qw(tempfile);
use CGI::Untaint;

my $handler = CGI::Untaint->new($q->Vars);

my ($fh,$name) = tempfile( "fileXXXXXXXX.jpg", DIR => '/home/web/user1
+/temp' );

my $url = $handler->extract(-as_url => 'set');

#my $url = $q->param('set');
#$url =~ m!^(http://[-.\w?&;]+.jpg)!
#  or die "The url does not look valid";
#$url = $1;

getstore($url,"../temp/$filename");
[download]

Please note that your script as written can be used to attack other webservers. You will want to restrict your script so it cannot be used by other people.

Update: Changed the code to use CGI::Untaint::url, which I trust more than my half-baked, restricted attempt

#!/usr/bin/perl
BEGIN {
    $|=1;
[download]

    print "Content-type: text/html\n\n";

    use CGI::Carp('fatalsToBrowser');
}

use CGI;
use LWP::Simple;
[download]

use strict;
use warnings;
[download]

$q= new CGI;
$my_data=$q->param('set');
$lower=1000; 
$upper=2000000; 
$random = int(rand( $upper-$lower+1 ) ) + $lower; 
$filename="file$random.jpg";
getstore($my_data,"../temp/$filename");
[download]

perldoc -q temporary

However chances are you're having an OS problem, not a Perl one. You say that the program doesn't work, but fail to explicitly mention what kind of error it gives you. I suggest you to do your best to help knowledgeable monks here to help you...