How do you fix your tablet if Google Play Services is stopping 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