Can't start system_health extended events session
First, identify the folder that is the problem. In SSMS, with results set to grid, run the following query:

SELECT target_data = CONVERT(xml, target_data) 
  FROM sys.dm_xe_session_targets
  WHERE target_package_guid = '5B2DA06D-898A-43C8-9309-39BBBE93EBBD'
  AND target_name = N'event_file';


Should return one row, and you should be able to click on it from the grid result. This will open a new tab with an XML document, which should look something like this:

&lt;EventFileTarget truncated="0"&gt;
  &lt;Buffers logged="0" dropped="0" /&gt;
  &lt;File name="C:\...path_to_xe_files...\system_health_0_131278486394000000.xel" /&gt;
&lt;/EventFileTarget&gt;


If the file name does not include a path component, the Extended Events default file location can be determined by looking in the system registry.  This code should do that:

EXEC master.sys.xp_instance_regread
       @rootkey      = N'HKEY_LOCAL_MACHINE'
       , @key          = N'SOFTWARE\Microsoft\MSSQLServer\CPE'
       , @value_name   = N'ErrorDumpDir';


Once you have the path where the Extended Events files are being saved, try to do something like this:

BACKUP DATABASE model TO DISK = 'C:\...path_to_xe_files...\model_trash.bak'
  WITH INIT;


This should give you a better idea of what the problem is - could be permissions, could be that the folder doesn't exist (maybe the configuration for the instance was moved), could be a whole slew of things. But this should point you in the right direction and get you a better error message.