NoMachine has this capability. You can run individual applications, disconnect, and reconnect them on a different X session or computer. I even set up the web based client that allowed me to tunnel into my running session over an HTTP proxy via SSH without installing any application.
This is an FOSS Dropbox that functions on Windows, OS X, and Linux. I have the server running on Ubuntu based off of http://www.x2b4.com/howto/how-to-install-ifolder-on-ubuntu-server/ and it works wonderfully. All of my files are synced immediately and it's open source and it's MY storage. No limitations, I pay for the bandwidth and electricity already, and now I get the benefit. Everything can be individually encrypted, if you want to offer the same service to family or friends. It's just a great solution. If you want to use Amazon, I'm sure you could just set it up to store the data on your Jungle Disk, or you can just switch to backing up your local data directory to your Jungle Disk.
I agree that DSL is probably the way to go. Your other option, since you have some embedded experience, is Angström. Build the qemu x86 image and extract it. It is designed for that size of ram.
'I second this. This is the FASTEST way to input information into a computer. Check with your school, sometimes there are license servers that you can hook up to. It takes maybe a week and you learn all of the "esc-s-esc" (sigma), "esc-int-esc" (Integral), ctrl-6 does superscript, ctrl-5 or 7 I don't remember does underscore, etc. All of it is immediately visible so you never make syntax errors and you never have to take your hands off of the keyboard. You also can do the computations real time and correct all of your prof's mistakes.