I have four computers at my work office and two at home. It's becoming a real hassle to keep my files organized and backed up. I would like to have a hard drive on one of my computers, and have all the other computers map to it over the internet.
It is very important that the mapped drive look and act like a regular drive with a letter from all of my computers.
Does anyone know if this is possible and how to do it?
I know there are services where I can backup my files online, but I'm concerned about security and privacy. I don't know where my files are or who has access to them.

I'm not sure about the drive letter assignment but what you are talking about is VPN access to a shared drive. Try this: https://secure.logmein.com/products/hamachi/vpn.asp?lang=en

If you are only using windows and mac operating systems, you may want to look at using Microsoft's Folder Share. It doesn't store/backup your data online anywhere, but syncs folders from multiple computers.

The service only works when your computers are online - it creates a private peer-to-peer network for your files.

https://www.foldershare.com/

VPN access to your network provides access to your files and folders as if your were physically connected to your network.

If you do not have a server in your office, then I would check into one of the inexpensive Network Attached Storage devices. They will have basic server functionality such as the ability to create shared folders.

If you have the Microsoft Small Business Server (SBS), it includes the Micorosoft Internet Security and Acceleration Server (ISA) which has the secured VPN functionality.

There used to be a service called XDrive that allowed this. But I'm not sure if its still operating.

You might consider getting a device like HP's MediaSmart server. That allows access to file from anywhere, but I don't think it works as a mapped drive.