In the poem "The Seafarer," is the speaker fully at home on land, on the sea, or in neither place.

This is a trick question because one could make a "correct" argument for any of the three possibilities.  In my opinion, however, the best answer to your question is that the speaker of "The Seafarer" is not fully at home in either place.  Let's look at the text for support here. 

First, the speaker continually refers to the sea as a "home," but one that is not his own.  He refers to the ocean first as he is called over "the horizon, seeking foreigners' homes" (38).  Just a little while later, the same speaker refers to the sea as the "whale's home" (60). 

Although never referred to as a "home," the sea is constantly referred to as a pull to the speaker's soul: a catalyst to spirituality.  The speaker says that "my soul / Calls me eagerly out" to the sea (36-37) or that "my soul roams with the sea."  This is incredibly significant in proving that the speaker's home can be found neither in the sea nor on the land.  Why?  Because the seafarer's home is in Heaven.  The speaker spends a considerable amount of time discussing the wrath of God and the fact that one cannot take material wealth into the afterlife.  Then, the speaker uses the word "home" one final time, proving my point.  The author reveals it in some of the last lines of the poem:

Our thoughts should turn to where our home is, / Consider the ways of coming there, / Then strive for sure permission for us / To rise to that eternal joy, / That life born in the love of God / and the hope of Heaven. (117-122)

Then the speaker uses the most vivid clincher sentence that anyone has ever used to end a literary work with a bang:  "Amen."