You need to set permissions on the directory Zope uses to store its data. This will normally be the var directory in the instance home. Zope needs to read and write data to this directory. Before running Zope you should ensure that you give adequate permissions to this directory for the userid Zope will run under.
Depending on how you choose to run Zope you will need to give different permissions to the directory. If you use Zope with an existing web server, it will probably run Zope as ‘nobody’. In this case ‘nobody’ needs read and write permissions to the var directory.
If you change the way you run Zope, you may need to modify the permissions of the directory and the files in it to allow Zope to read and write under its changed userid.