To be specific, Cloud is a vast term, including cloud computing, cloud storage etc. To be more specific you are looking for a solution to run your own private cloud i.e file sharing, storing on a server & you don't want to pay to hosting companies for storage, Domain Name Server or for software/app for accessing your files on your cloud.
Ok i have a good news for you u can do all of that, but u need to buy a router with a USB port having functionality of using that USB port as FTP server storage. You can check Asus website, router section for more info.
After buying a USB FTP router you can run your own FTP server without the need of a computer to be powered on all the time as the router can run the ftp server on its own firmware & processing power.
Next major problem is, home users generally do not have static ip-address, if you have a static IP you are good to go but if your ISP gives you dynamic IP address, u need to register for a free Dynamic DNS service from http://www.noip.com/ this will solve the dynamic ISP problem. (this is important because running a ftp server on router with dynamic IP means the IP-address will change every time you switch off router, This means you have to manually check the IP address of your FTP server every time you switch on your router, to avoid this use noip.com)
Finally to transfer files from you cloud (your ftp server running on your router with storage of USB/USB-hardrive plugged in on USB port behind the router), you can
1)Use free FTP file transfer software's like filezilla.
2)Map a network drive on your computer, pointing the drive to your FTP server Using free DNS service from "noip.com" you registered for.
3)Use online file transfer FTP websites to transfer file to/from your cloud FTP server.
4)Use browser based FTP clients like FireFTP add-on for Firefox.
AS you have updated your original post ,laying stress on folder sync functionality. I recommended a ASUS router, which provide smart sync, from there free app called "ASUS AI cloud " , I think it is exactly what you are looking for, have a look here http://event.asus.com/2012/nw/aicloud/
You can give exclusive user rights to people accessing your storage drive on your cloud FTP server by logging in your router page as admin. This is how you can run a basic private cloud like google drive, dropbox, skydrive etc from your home, without paying anybody.