Eanna means the house of heaven, and by the historical period it is unambiguously the sanctuary of Inanna at Uruk, the goddess of sexual love, of war and of the morning and evening star. Her temple there is the fixed point of a good deal of Sumerian literature. In the poem of her descent to the underworld she abandons her temples one by one, Eanna among them, to go down. In the great hymn attributed to Enheduanna, the priestess who served at Ur, Inanna is praised in terms that pull the whole pantheon into her orbit, and the political weight of her cult at Uruk stands behind it. Anu, the sky god, is also housed at Uruk, and the relationship between the two cults on the same site changed over the millennia, with Anu prominent again in the Seleucid period.
What cannot be settled is the beginning. The monumental buildings of the earliest levels are enormous, elaborately decorated with cone mosaics, and repeatedly demolished and rebuilt, and no readable text from those levels says who they were for. Reading Inanna back into them is an inference from later certainty, not a reading of evidence, and the atlas records it as an open question with the study that could not close it attached.
The ruin field is protected as part of a World Heritage property inscribed in 2016, which groups Uruk with Ur and Eridu and with the marshes of the southern plain. Nothing is worshipped there now.