Mortal coil is a poetic term that means the troubles of daily life and the strife and suffering of the world. It is used in the sense of a burden to be carried or abandoned, most famously in the phrase "shuffle[d] off this mortal coil" from the "To be, or not to be" monologue in Shakespeare's Hamlet.
Derived from 16th Century English, "coil" refers to tumults or troubles. Used idiomatically, the phrase means "the bustle and turmoil of this mortal life."  "Coil" has an unusual etymological history. It was coined repeatedly; at one time people used it as a verb to mean "to cull," "to thrash," "to lay in rings or spirals," "to turn," "to mound hay" and "to stir." As a noun it has meant "a selection," "a spiral," "the breech of a gun," "a mound of hay", "a pen for hens", and "noisy disturbance, fuss, ado." It is in this last sense, which became popular in the 16th century, that Shakespeare used the word.
"Mortal coil"—along with "the slings and arrows of outrageous fortune," "to sleep, perchance to dream" and "ay, there’s the rub"—is part of Hamlet’s famous "To be or not to be" speech. "Coil" is no longer used as a synonym for "disturbance."
Should there not have been originally 'shuttled off'? This verb itself no longer exists but 'shuttle' is an implement used in weaving. Accordingly, the meaning might be: 'when we have unwound and worked off this coil of mortality.'
In this way, the length of our life is metaphorically the length of thread that is coiled on a spool, a metaphor related to the ancient Greek mythological figures of the Fates. As we live, the thread is unwound from the coil by the shuttle of the loom of time.
- Oxford English Dictionary 1979 edition