Sounds to me like you don't have the necessary permissions to save the game.
Go to SOMA's steam folder (C:/Program Files (x86)/Steam/steamapps/common/SOMA/), right click Soma.exe and go to its properties and tick "Run this program as an administrator" under the Compatibility tab.
If it is a permissions error, the above should fix it.
If you're lucky, you may be able to recover some of your saves by doing the following steps.
- Hold the Windows Key and press R. This will bring up the "Run" window.
- Excluding the quotes but keep the percentages, type "%localappdata%" This will bring up an Explorer window.
- Scroll down to the VirtualStore folder.
- Your saves will be located somewhere in there. Likely in a directory like "Documents/My Games/SOMA/Main/<steam_user>/"
Saves are stored at C:\Users\<Username>\Documents\My Games\Soma\Main\<steam_user>\
If they are in the VirtualStore folder, copy the saves into your <steam_user> folder.