This blog post describes how to install and activate the sandbox solution which adds an anonymous access ribbon button to all lists. By the use of the button, you will be able to enabled access to list for anonymous users on Office 365.
Activation: http://www.fiechter.eu/Lists/Posts/Post.aspx?ID=13&mobile=0
Download: http://www.fiechter.eu/Solutions/Wsp365.Anonymous.zip
Update 24.04.2013: Newer Office 365 tenants have to use: http://www.fiechter.eu/Solutions/Wsp365.Anonymous.2013.zip
Activate the feature: http://www.fiechter.eu/Lists/Posts/Post.aspx?ID=11&mobile=0