Google Play Store not working

Hello,
For the last 2 hours my google play store isn't working. The problem is it suddenly started showing the message " Google Play Services keeps stopping" and although it shows the little cross to close the app, it clearly doesn't work.
Before even looking online for any solutions I tried to troubleshoot it with: clearing cache, uninstall updates, even unistall app (which doesn't allow me etc. After searching online for some troubleshooting, I found out that I had already gone through some of the steps, I still repeated them. For info, my Play Store doesn't work either. On one of them it says to remove my Google account but that means removing all important data like messages and contacts and more.
I need advice and guidance. I appreciate your help on this.

Sylvia

Fyi, when I go to "Accounts and backup" and "Accounts" next and then choose my Gmail account and try to remove it, it doesn't do anything. I click on the button "Remove account" in order to reset it and...nothing. Are there any other options before a factory reset?