No, no. Not that type of anonymous access. I’m sure this is of no interest to them.
When using an Office 365 public site and you create a document library or list, by default you have to be logged in to view it. Not great if you want to publish a bunch of documents out to the public…anonymously.
The solution
1. Download this sandbox code solution from codeplex: http://anonymous365.codeplex.com/ and unzip it.
2. Go to https://yourpublicsite.sharepoint.com/_catalogs/solutions/Forms/AllItems.aspx where “yourpublicsite” gets replaced with your site name and upload the WSP file.
This will be get you to the hidden site collection solution package upload area. There is no direct navigation.
3. Activate the solution
4. Go to your Office 365 SharePoint admin site and allocate some server resource – by default public sites have 0 (sandboxed code) resources so you’ll just get an error if you try to use it before giving it some points.
URL if you’d prefer: http://yoursharepoint-admin.sharepoint.com/_layouts/15/online/SiteCollections.aspx
5. Put a check in the box next to your public site and click on “Server Resource Quota”.
6. In the dialogue that pops up, allocate 300 server resource points to the site (should be more than enough) and click on save.
7. Once you’re done you can go over to your document library and configure anonymous access to your library.
Once you’ve done these steps your library should be available anonymously.
References:
http://365tutor.org/enable-anonymous-access-in-office-365-public-sites
http://anonymous365.codeplex.com/
http://community.office365.com/en-us/f/148/t/193899.aspx