There are multiple settings in Tuesdays With Morrie, the most prominent being Morrie's home as well as the university as it appears in Mitch's flashbacks. As the book moves forward, Morrie's room moves to the forefront, becoming the main physical location where the events of the book take place. In that sense, the setting is specified pretty clearly.
However--moving on to the second part of your question--I wouldn't call the physical setting "fully described." Albom leaves much to the imagination as far as filling in the gaps of a typical home goes. We get sparse details here and there, like hardwood floors, wicker chairs, and the persistent but undersized hibiscus plant that lives in Morrie's room, but the rest is not developed, because there's actually a much richer metaphorical setting that Albom does take great care and detail in describing.
The real setting of the story is Morrie himself, and his deteriorating body. We get this in agonizing detail; for example, in the "Fourth Tuesday" section:
I noticed that he quivered now when he moved his hands. His glasses hung around his neck, and when he lifted them to his eyes, they slid around his temples, as if he were trying to put them on someone else in the dark (77).
Morrie's physical condition worsens continually throughout the book, making his body the actual stage upon which all the drama of the story occurs. The reader watches in horror as Morrie slowly looses control of his muscles and drowns while his lungs fill with fluid. But it's that impossible terrain that makes Morrie and Mitch's spiritual journey so significant. Despite overwhelming potential for defeat, depression, and resentment, these two men stubbornly continue their Tuesday lessons with the goal of preserving Morrie's legacy of wisdom, trying to come to peace with who they are and, above all, determining how to live.
It is incredible that these vast, earth-shattering things are happening in such a non-remarkable, sparsely detailed setting, which is really just a room like any other. But it makes sense when you consider the message of the text, which is that physical things (even our own bodies), are of little consequence in comparison to the great potential that exists in the human mind and spirit.
What she said!