There are two parts to the Rag and Bone Man quest. The bones needed for the first part are easier to obtain compared to the second part. The second part is a list of bones posted on the wall of the Odd Old Man's hut. The names of the bones in this Wish List will be taken off as they are turned over to him. This part of the quest is no longer needed for the Quest Cape.
Go east of Varrock, through the gate near the earth altar as if you were travelling to Canifis (north of Digsite) and half way there, you will find the Odd Old Man in the limestone quarry to the north.
Talk to him, and he will tell you that he needs several bones to complete his collection.
Once you have collected the bones, go to Draynor Village and talk to Fortunato (found by the market square) there. He will be able to sell you some jugs of vinegar for 1 coin each. Note: Do not steal from the market stall or he will not talk to you (he will if you have a vial of stench (a)). After you buy 8 Jugs of vinegar talk to the Odd Old man for further instructions. *Note: You can buy the Jugs of vinegar and add it to the pot without talking to the Odd Old Man for instructions.
Use a jug of vinegar with a pot to get a pot of vinegar, and then use the bone with the pot of vinegar to get a bone in vinegar. Repeat for all 8 bones.
NOTE: Be careful that you do not accidentally "empty" a bone in vinegar, because if you do, you will lose the bone and have to collect that bone again.
Go back to the old man taking the 8 bones-in-vinegar, and 8 logs. Or alternatively, bring a hatchet; there are trees near the Odd Old Man at the bottom of the stairs where logs can be cut. Lighting the logs once will polish the bones.
To polish the bones, click the pot boiler located next to the Odd Old Man to add logs, click it again to add a pot with a bone-in-vinegar. (you will not receive XP for lighting logs). It takes 12 seconds for each pot to boil. Repeat this for all 8 bones.
Rag-and-bone man is a British phrase for a junk dealer. Historically, the phrase referred to an individual who would travel the streets of a city with a horsedrawn cart, and would collect old rags for converting into fabric and paper, and bones for making glue, scrap iron and other items. They often traded them for other items of limited value.
Even more evidence for another Rag and Bone Man quest was found in Postbag 41. One of the sheets of paper in the Chaos Elemental's suitcase was titled "MENTOR A BANDWAGON." This unscrambles to form "RAG AND BONE MAN TWO." In addition, the bottom of the paper had the words "No 156 in the world." At the time of release, there were 155 quests, and Fur 'n' Seek was released as the 156th quest.
Jagex hinted in their weekly Twitter hint from 7-8-09 that the next quest would be Rag and Bone Man 2. It was released as Fur 'n' Seek on 10 August 2009.
If the quest is completed, your Adventurer's Log will state "The Odd Old Man wanted some bones. Or was it his bonesack? Will we ever know?"
Even when using a Dwarven Army Axe, the animation used for lighting the logs uses a Tinder Box.
Upon completion of this quest, the player tells Juna "...I fetched all the bones the Odd Old Man wanted and he gave me a reward!" to which she express puzzlement about the Rag and Bone Man by replying "What could that man have wanted all those bones for? Collecting that many bones cannot be a balanced activity."