Solution
If you just have save error 0 you can try if you can save by using save as and save to a new name. If you have both issues, follow instructions below:
Close the game
Go to Documents/Electronic Arts/The Sims 4/cache
Delete all the cache files.
Move your saves folder to your desktop
Start the game to create a new saves folder
Your game should now be back in working order but your existing save games are also gone.
Try to put each save game from the folder on your desktop into the new one (one at the time) and see if any of them still work. The one with the name slot001 is the auto save and is most probably the culprit since they tend to corrupt when the game crashes.
The game keeps the last few saves you make. Here’s instructions how to recover them: How do I restore a backup ?
If not, I’m afraid that you will need to start over.