Is this a 360 or a regular xbox, your varied between subject and body. If a regular Xbox 360, then yes it will work. The kinect units come with USB plugins, so if you don't have a newer 360 with a kinect plug, it will work just fine with older units as long as you have an available usb port.
On a technical sid, the newer 360's Kinect plug carries USB and power in one plug. But for older X360 units without that plug, the kinect comes with a USB power plug Y cable. So you'll have a USB cable to your xbox 360, and then another cable that goes to a wall outlet to power it.
20GB of space is enough, the only requirement I've seen is that you at least have a hard drive, as internal storage and 512MB cards are not enough to hold things.