The tale starts the day after Anodos' twenty-first birthday. He discovers an ancient fairy lady (whom he learns to be his grandmother) in the desk which he opens with a key that he inherited as a birthright from his late father. After the fairy shows him Fairy Land in a vision, Anodos awakes the next day to find that his room, crafted after natural elements, is taking literal form and transforming into a wood. He discovers that he has been transported to Fairy Land.
Anodos then encounters a woman and her daughter in a cottage who warn him about the Ash Tree and the Alder Tree, who seek to destroy him. He is told that the spirits of these trees can leave their tree-hosts and wander throughout Fairy Land. He then explores the world of the fairies, which live in flowers, causing them to glow. The flowers, he is told, die if the fairies leave. He then has a nightmarish encounter with the spirit of the Ash Tree, escapes, and finds rest in the warmth and love of the Beech Tree's spirit.
After this, he finds the statue (fondly called "my Marble Lady" by Anodos) by Pygmalion. After he sings to it, the statue flees from him. He pursues the lady and finds a woman he believes to be her. However, this lady is actually the Maid of the Alder Tree in disguise. The spirit of the Ash Tree joins the Maid and is close to killing Anodos when he is saved by Sir Percivale (who chopped the actual ash tree with an axe). Anodos then meets a woman and her daughter who believe in fairy tales and the magic of Fairy Land, despite the disbelief of the woman's husband. Anodos also finds his shadow, an evil presence that follows and torments Anodos throughout the rest of the story.
Anodos finds a palace that mysteriously belongs to him, and it contains a room with an inscription that reads "Sir Anodos." In the palace, he reads the story of Cosmo of Prague. Cosmo is a believer in fantasy who sacrifices his life to free the soul of his lover from an enchanted mirror (whether the event was a fictional story made by an author from Fairy Land or if it was a recording from an event in Anodos' world is left ambiguous).
Anodos spends much time in the palace, relating his various wanderings and readings. In one such wandering, he comes upon corridors filled with still statues. Hearing the last vestiges of song from the corridors, and considering the statues as recently frozen into immobility upon his approach, Anodos ventures deeper and deeper into the halls. He dreams of the marble lady, that she alone has an empty pedestal among the statues. He later finds this pedestal, and, figuring a way in which to trick the statues into continuing to dance as he enters the room, he eventually sings to the pedestal. The marble lady materializes, but Anodos attempts to grab her. She flees and disappears. Anodos follows, going down into a strange subterranean world with gnome-like creatures (like the German Kobolds) that mock him.
Anodos escapes this place and finds himself in a stormy sea. When a boat arrives, he boards it. It takes him to an "island" with a cottage with four doors which is inhabited by an ancient lady with young eyes. Anodos enters each door in turn, each containing a different world. In the first he becomes a child again, remembering the death of his brother. He comes back to the cottage crying. In the next door he finds the marble lady and Sir Percivale, alive, well, and in love. They are talking about him, and Anodos (previously unnoticed) makes a last outburst of his love for the marble lady. They leave, as does Anodos. The next door recounts the death of a loved one of Anodos, and he finds his family mausoleum. His ancestors help him back to the cottage. Finally, Anodos travels through the last door ("the door of the timeless") but is saved by the ancient lady without remembering anything. The ancient lady says that because she saved him, he must leave (the "island" in fact has an isthmus).
Next Anodos finds himself with two brothers who also call Anodos their "brother," due to a prophecy given to them that a third would come to help them. They are forging armor and swords in order to fight three giants that have fortified a castle nearby, to the dismay of the townsfolk. The brothers are the sons of the king. Anodos joins them in their fight, but they are attacked unarmed by the giants. The brothers die, but Anodos lives, becoming a hero of the kingdom. He wanders to tell a woman, whom one of the brothers loved, of his honorable death, but he finds instead a manifestation of his shadow, who imprisons Anodos in a tower. Anodos is saved by the song of a woman whom he had met before in fairy land, and he is not troubled by his shadow ever again. Anodos becomes the dedicated squire of the knight, and they become good friends. They come upon a temple full of worshipers doing an unknown evil to a select few. Sir Percivale, always seeing good in people, is deceived, but Anodos rises to end the practice. He destroys the idol made of rotting wood that is sitting on a throne. He is killed by the multitude before Percivale can save him. In death Anodos finds peace, having died nobly. He floats, overlooking things, and finally awakes alive in the "real" world, never forgetting his experiences in Fairy Land. His sisters inform him he had been gone 21 days, but to him it felt like 21 years.

Question: Who is Anodos obsessed with?
Answer: The Marble Lady
Alternative answer: the Marble Lady statue