Yes, between Public and Mass Transit, it is doable.
Public Transit includes the OCTA Bus System, the LA Metro Bus system, the ART local shuttle system, plus both Amtrak and Metrolink Trains.
Mass Transit includes Gray Line Anaheim, Starline Tours, and things like Shopping Malls running their own shuttles for a fee.
There are a couple of Top Questions here at the Anaheim Forum that discuss how to get around without a rental vehicle.
Many folks come and stay in the Anaheim area for a couple of weeks and don't bother getting a rental car.
Some decide to split up their stay, staying in the Anaheim area, and then opting to stay in San Diego and the Hollywood area of Los Angeles, which also have good access to public/mass transit.
Edited: 11:58 am, February 23, 2013