It is possible to explore the downtown area on foot, but that's worth, IMHO, a half a day at most. There's so much to see outside of Anchorage (even just a few miles away) that you're not really doing the area justice by staying in town only.
The only touristy thing really worth seeing in Anchorage outside of the downtown area is the Alaska Native Heritage Center, and I'd bet they probably run shuttles between there and downtown.