A really good but expensive alternative would be running Citrix / Teamviewer / etc but then you would need two factor authentication and even then you would have the issue of full write access by the client operating system.

Good idea!

OP, have a computer somewhere that has your own Windows (or Linux, of course) on it, with RDP enabled and connect to that. Assuming your internet speed is good enough to make this comfortable.

On your home PC:

• Go to System
  • Click Start (or hit the Start button on keyboard)
  • type "System"
  • Hit Enter when "System" shows up in the Start Menu (or use arrow keys if it's not the first result
• On the left, click "Remote settings"
• Click "Allow connections only from computers running Remote Desktop with Network Level Authentication (more secure)".
• That option enables RDP, but locks out XP-era RDP clients, which should be no problem.
• Forward port 3389 from the internet to your own computer using your home router (see router manual, learn to do this, it's incredibly useful).
• Find your home IP by asking Google: "What is my ip?"

When using a public machine, assuming it's using Windows:

• Click Start (or use keyboard Start button)
• Type "remo"
• Check that "Remote Desktop Connection" is the first search result in the start menu, if not, use arrow keys to select it
• Hit Enter, the program now opens.
• Enter the IP address found in the last step from the list above
• Enter the username and password you use on your home computer
• DO NOT SAVE your username and password on the public PC.

You now have a connection to your own computer, with all its documents and programs. Nothing will be saved on the public PC, except some image cache at most. The public PC isn't running programs, it only display the graphics it gets sent from your home PC and sends mouse/keyboard back to that machine.