This is pretty similar to Content Deliveries in classic, but public links let you share documents easily via links, but without password protection, expiration, or usage tracking.
Here are a few things you need to check to enable this:
1. Content Deliveries and Public Links
Make sure you see this menu in setup page, otherwise reach out to Salesforce support to enable it.
Once you in Content Deliveries and Public Links, enable this "Public Links can be enabled for users (Requires Content Deliveries)".
As you see from above this also requires Content Delivery.
2. Content Permissions
Search for Content Permissions under setup menu (until Spring '19 release - this only available in Classic, so you need to switch back to Classic). This will allow you to add, edit, and delete Library Permissions - a Library Permission is a set of user privileges for Salesforce CRM Content libraries.
Click Edit on Library Permission Name and enable Deliver Content.
If you do not see Deliver Content action, reach out to Salesforce support to enable it.
3. Create Library (optional)
This step only needed if you need to create a new library, make sure you have the appropriate permission to create Library.
4. Create Folder
Once you are in the library, click New Folder button to create a folder.
5. Create Public Link
Without correct setup and appropriate permission, the user will not able to click a public link for sharing.
Once all setup correctly, you will see Public Link option from the drop-down menu for the folder.
Note: if you delete and recreate the public link, the URL will be changed.
Once you uploaded files, here is the final result, the content accessed from the public link will be updated in real-time.
If you see a blank screen when open the public link, there is a known issue which will be fixed in Spring 19 release <-- update this has been fixed in Spring 19 release.
Reference: Share Multiple Files in a Folder Using Public Links