How do you keep Google Play Services from stopping or get it to start back up? It has deleted all my apps and is now telling me every time I try to download that I don't have enough space. What do I do?
maybe you really out of space! applications are installed in /data directory which is mostly 2gb try to move some of your apps to SD.
About your stopping go back to your Settings > Apps > Google play services, remove if they is any update and clear data and reboot, let the play update itself
I kept getting the message "Google whatever has quit working" in any of the Google Apps I tried to use. Here is how I fixed this problem.
I went to Applications in Settings, then went to "all" then scrolled down until I saw the Google applications begin and I stopped them and disabled them and dumped the cache in them all and uninstalled any updates if there was that option available.
Then I turned my tablet off and back on again and it instantly said "Google has stopped" so I reset the tablet by sticking a straight pin in the reset hole on the back of the tablet. I counted to 10 and then turned it back on and immediately went to the play store and started reinstalling everything just like it wasn't in my tablet at all before or I updated the apps one by one, whichever option was available. That fixed the problem.
Maybe I went above and beyond but it was easier than trying to figure out exactly which app was causing the issue so that's how I fixed it.