Yes u can download stuff on ur Xbox when it is "off". Its called background download. When ur done gaming but u still have something in ur Active Downloads, u can turn off the Xbox but it will not completely shut off, instead it will finish downloading ur game or whatever. U can activate the Background Download by going to Settings, System Settings, Console Settings, Startup and Shutdown, and select Background Download to turn it on. The on/off button will blink when u turn off the console and u have something downloading, and will stop when the download is completed and the console will turn off completely.