It doesn’t exactly trip off the tongue, which may have contributed to its rarity in modern English. We also have other words to express the same idea, such as indescribable and inexpressible. Some writers have chosen it as an alternative to ineffable, something that’s unutterable, too great for description in words:
Shall I not crown them with immortal praise
Whom I have loved, who have given me, dared with me
High secrets, and in darkness knelt to see
The inenarrable godhead of delight?
The Great Lover, by Rupert Brooke, 1914.
It’s ultimately from Latin enarrare, to recount or describe fully, whose root has bequeathed us narrate and its relatives. Classical Latin scholars attached the negative prefix in- to make the adjective inenarrabilis for something that’s inexpressible or which defies description. In the fifteenth century, the adjective inenarrable based on this arrived in English via French.
More prosaically, some authors have borrowed it to demonstrate the breadth of their vocabularies or to empurple their prose, in this case as a sarcastic counterpoint to the inarticulate character in question:
A long draught of the corrosive nectar, to be savoured with the inenarrable contentment which the divine fruit of such a pilgrimage deserved, washed gratifyingly around Mr Uniatz’s atrophied taste buds, flowed past his tonsils like Elysian vitriol, and swilled into his stomach with the comforting tang of boiling acid. He liked it.
The Saint in Miami, by Leslie Charteris, 1941.