The site acts as the program, you can receive and trade coins without needing a program.
I personally may start using this site for receiving money and once I get enough I'll just move it to my wallet (the bitcoin program).
As far as I can tell this site doesn't require you to use any program.
If you were to say run the bitcoin program once or twice to create the wallet.dat file; you could use the bellow method to secure your coins and then you could just remove the program later.
https://en.bitcoin.it/wiki/Securing_your_wallet#Making_a_secure_workspaceAlso, how is it possible that you can't spend upwards of 50mb of space? Worst case, just run it off of a USB.