What are permissions are needed to upload a file via a public Salesforce community page?

UPDATE

The component is now supported for guest users with Spring '19.

• Let Community Guest Users Upload Files release notes.

  Promote file sharing in your communities. Let guest users upload files using the lightning:fileUpload component.

• From documentation for the component:

  By default, guest users can’t upload files. You can enable the org preference Allow site guest users to upload files.

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This section was relevant in Winter '19, removed now as the support is now available with Spring '19.

Based on the latest updated question.

lightning:fileUpload is not supported for Guest Users. Refer to the below excerpt from documentation.

Guest users can't add files to Communities.

Your only option here may be to utilize a VF page (as mentioned by Pranay in the other answer)

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The documentation for lightning:fileUpload mentions the following:

File uploads are always associated to a record, hence the recordId attribute is required

And based on the help article, if you need to attach a file to a record, you will at least need a READ access on the object.

Attempt 1:

As it was not possible via profiles, I tried doing it via permission set.

I created a new permission set and added below permission for files

1. Create Content Deliveries
2. Create Public Links
3. Moderate Communities Files

And when I tried to assign that permission set to the guest site user, here is the error I get.

It didn't let me assign permission set till I removed all the above 3 permissions.

Can't save permission set AA, which is assigned to a user with user license Guest User License. The user license doesn't allow the permission: Create Content Deliveries
Can't save permission set AA, which is assigned to a user with user license Guest User License. The user license doesn't allow the permission: Create Public Links 
Can't save permission set AA, which is assigned to a user with user license Guest User License. The user license doesn't allow the permission: Moderate Communities Files

Salesforce is not willing to allow Files via guest user context and hence it will fail with all the standard methods possible.

Attempt 2:: If I cannot use lightning, let me revert back to visualforce page.

I created a Page and VF controller:

<apex:page controller="ContentController">
<apex:form>
    <apex:inputFile value="{!file}" />
    <apex:commandbutton action="{!upload}" value="Upload" />
</apex:form>
</apex:page>

Controller:

public class ContentController {
    public blob file { get; set; }

    public PageReference upload() {
        ContentVersion v = new ContentVersion();
        v.versionData = file;
        v.title = 'testing upload';
        v.pathOnClient ='/somepath.txt';
        insert v;
        return new PageReference('/' + v.id);
    }
}

And then used my Community builder to drag and drop that VF page on my Guest access site. I assigned VF page and controller access via Guest User Profile. I published it and then tested in an incognito window

And voila, I was able to upload file and generate an ID for it using visualforce and Apex.

Only Drawback: Less than 6MB can only be uploaded. Happy days.