@Marinarn - wrong response. This is someone looking to secure their current server rather than someone looking to purchase a NEW unmanaged server.
@wackter - the file manager program that you were using is possibly using a shell like script to bypass normal system protections. Can you PM me the file script you're using?
If you have suphp loaded on your server, then you should be ok, and this shouldn't be happening. Also, check with your script to see if you're able to WRITE to one of those outside directories, and are able to traverce through the folders. I've seen a lot of VPS machines (and dedicated servers) in operation with hosting companies that expose all client usernames by allowing a simple script to execute.
Verify that the script is not loading into the /tmp folder on your server as if it can load into there, you can read through the server using remote call scripts. Make sure that the /tmp is secured from execution of files.
You may also want to check out somewhere like
www.platinumservermanagement.com or similar 3rd party management company who can get in and help diagnose quickly and cheaply. I've worked with that company about 10 years ago and they are still a highly recommended company.
Are you running cPanel on your server, or another control panel?