Always wondered that 'cause they look so well drawn in high definition and it'd be really a shame if there was no way to actually preserve those background pictures used on the Xbox 360 Store. I suppose acquiring it via capture card would be a solution but isn't there a lossless one that keeps it in its original quality?
You might be able to packet sniff where the 360 downloads the images from: Packet Sniff Xbox 360 Network Traffic & Dashboard Updates with Wireshark then go download the images directly on a PC. That would require a bit of time investment, however if you can get WireShark working then it's just as case of capturing packets and browsing the 360 dashboard as normal, letting the images download. After review the packets you captured and see if you have any "GET" requests for the images, then re create the URL as shown in the tutorial above to download the images using your web browser on a PC. It might well be easier and quicker to use a capture card though.