This is very simple: you seem to not have enough hard drive space for a system update that your console is trying to install.
Error 8007 0070 occurs when you install an update or buy content on...
(I've pasted some bullet points for you)
Error 8007-0070 occurs when you install an update or buy content on Xbox 360
You may see the following error code and message when you try to update your Xbox 360 console or purchase content:
The "x" characters in the error code vary.
This may mean that there's not enough space on your storage device (mostly likely the Xbox 360 hard disk) to install the update or the content.
A console update or content download may require significant hard disk space. If you don't have enough free space, you need to move or delete some items."
You can probably get around this by using a 32gb USB thumb drive and moving some of your content that's hogging up your console's space onto that thumb drive temporarily.
Heres the Xbox.com info to get you started about moving and deleting hdd space--which also has info about using a USB thumb drive, too. Copy move or delete Xbox saved games profiles and avatar items