Although the "great pain" Dickinson refers to is ambiguous--it could be the pain of losing a loved one to death or the loss of love or friendship--she accurately describes how someone reacts to a very recent traumatizing loss.
Dickinson's metaphors create not only the atmosphere of loss but also explore, in a strikingly modern way, how the mind and body behave during the initial part of the grief process:
The nerves sit ceremonious like Tombs--/The stiff Heart questions. . . ./The Feet, mechanical, go round. . . . A Quartz movement, like a stone.
Essentially, the faculties, both mental and physical, shut down: nerves no longer carry their messages to various parts of the body; the mourner's heart feels no emotion; the body's movements are actuated as if they were part of a clock rather than a purposeful intelligence. Dickinson has accurately detailed one important stage of grief that allows a human being to exist in the face of great emotional trauma without the brain and emotions being engaged.
The mood--created by "Tombs," "stiff Heart," "mechanical," "Wooden," and "Quartz"--is somber and, more important, expresses the numbness of mental and spiritual detachment. In one sense, this poem is strikingly modern because Dickinson correctly describes how we often react to overwhelming grief--a defense mechanism allows us to carry on daily activities without conscious effort.
The last stanza carries the overall metaphor of unemotional grief one step farther by accurately describing how people freeze to death, another unusually modern touch:
As Freezing persons, recollect the Snow--/First--Chill--then Stupor--then the letting go--
We know that people who die by freezing experience painful cold during the first stage and then feel comfortable just before they die, and Dickinson compares a grief-stricken person's analogous reaction to loss--a grieving person feels intense pain, then numbness of body and mind, and then "the letting go." Because the mind and body can operate like a clock, however, without the intervention of conscious effort, the grief-stricken person is able to survive.