The Door of Loathing is one of three doors in the Necrohol of Nabudis that can be opened only with a special medallion. Behind the door waits a small, but powerful, hare-like monster that must be killed to fully explore the area. After this monster and the monster behind the Door of Horrors are destroyed, the Lusterless Medallion transforms into the Medallion of Might, which allows you to open the Door of Despair and fight the esper Chaos.
Speak to Roh'kenmou at Old Dalan's house in Lowtown. He will tell you a story about a medallion that was broken into four parts. Old Dalan proceeds to give you the first part of the medallion.
Visit the vacant house in the North Sprawl, and talk to Deeg. Read the letter on the table, and then talk to Deeg again. Neither Deeg nor the letter will be there if you have not completed the Waterway Haunting hunt, however.
Visit Sorbet the moogle at Westgate. Complete the Lost in the Pudding hunt, if you have not already done so, and Sorbet will give you the Sluice Gate Key.
Take the key to the central waterway control area in the Garamsythe Waterway, and open all the gates. Close gates 4 and 11, then open gate 11 and close gate 3. Next, open gate 4 and then look on the ground around gate 3. You will find the Dull Fragment on the ground.
Complete the Crime and Punishment hunt, to acquire the Stolen Articles. The petitioner is the contrite thief in the North Sprawl. Defeat the Orthros in the Garamsythe Waterway, and return the Stolen Articles to the thief. You will be rewarded with the Blackened Fragment.
Speak to Ma'kleou in the Nabreus Deadlands. Return to the South Sprawl, and speak to Filo. Visit the Southern Plaza in Rabanastre, and talk to the Curious Woman by the fountain. Go to the Muthru Bazaar, and speak to the two merchants there. Then go to the east end of Rabanastre, and visit Yugri's magick shop. Speak to the imperial soldier near the door, and tell him about the woman at the fountain. Go to the North Sprawl, and talk to Kytes (he is near Warehouse five). Then find Filo at the southern end of the South Sprawl. Ask to see the woman from the fountain. Return to the magick shop, and speak with the imperial agent again. He will give you the Grimy Fragment. You must talk to these characters in this order, to get the fragment.
Take the fragments to Roh'kenmou at Old Dalan's house, then travel to Charlotte's Magickery in Molberry. Roh'kenmou will be there. Talk to him, and then find Otto in the Alley of Muted Sighs. He gives you the Moonlight Medallion, which you must then return to Roh'kenmou.
Return to the Nabreus Deadlands, and talk to Ma'kleou again. Find the secret path at the right of the northwest corner of the Slumbermead, and follow the path until you reach a shrine. Ma'kleou will be there, and he will give you the Medallion of Bravery, the Medallion of Love and the Lusterless Medallion.
Return to the Necrohol of Nabudis, and place the Medallion of Love in the Door of Loathing in order to open it.