Just like I can remove access to a particular object, how can I remove the user's access to Files or Attachments? Right now they can still access it if they use the global search function. I think there is no way to disable the global search function too. Any ideas?
Who Can See My File?
Your files in Salesforce can be private to you, privately shared, or visible to your entire company. Learn how to identify a file’s sharing settings and how to change them. Check this doc for details
Make a File Private
You can restrict access to shared files that you own by making them private.Details here
From the file detail page, click either Share File Share File | Sharing Settings or Show All on the Shared With list.
In the Sharing Settings dialog box, click Make private (or Restrict access if the file originated in a Salesforce CRM Content library).
In the confirmation dialog box, click Make private (or Restrict access).
View Where a File Is Shared
Find out who can see a file and their level of access. From the File Detail page, you can access a full list of every person or object the file is shared with.
The Shared With list on a file detail page shows you who the file is shared with and their level of access. Check details here
Yes, unfortunately global search can't be disable for particular user or profile. Check this idea