I've clear cache on Play Store many times as well as clear cache/clear data which then routes me to a manage space button and/or a page with my multiple Google apps to manage spaces of app(s) of my choice that I'm trying to clear. I usually don't have to sign in again. Maybe because of dual sign ins on various Google apps however clearing data could very ...
There is absolutely nothing to worry about this issue. All you need to do is the following:
Find and Download Google Play Services APK (Pure APK File)
Copy the file in your mobile phone memory
Click on the APK file that you copied to your smart phone
After some time it will install Google Play Services on your mobile phone.