Start by talk to the Odd Old Man at the limestone mine between the Lumber Yard and the temple on the River Salve. He asks for help gathering bones 'for the Museum'. He asks for special bones from each of the following monsters:
- Giant Rat
- Big Frog
- Giant Bat
He also asks that you bathe and boil them in Vinegar before giving them to him.
Obtaining the Bones
From the Odd Old Man, follow the path heading west towards Varrock. Once you're through the gate, turn south and continue until you're near Varrock's southeastern mine. Here you will find the Giant Rat, Bear, and Unicorn. Kill them for their bones.
Note: They each drop unique bones that cannot be buried. The drop is not a guaranteed drop, but it does drop very frequently. You can only have one of each bone at a time.
Teleport to Lumbridge and head north towards the sheep pen. Kill a goblin and pick up the Goblin Skull. Then kill one of the rams mingling around the sheep for its skull as well.
Next, head into the Lumbridge swamp (with your light source, tinderbox, and a rope). Look for the Candle Seller near a dungeon entrance. Use a rope on the large tree in the hillside, and climb down. Make sure your spiny helmet is worn, then run west and down the south passage, then east through the narrow passage until you reach the Big Frogs.
Get its bone and head to Karamja, then kill a monkey in the jungle around Musa Point. On top of the volcano, enter its dungeon and kill a Giant Bat.
A Cask of...
At the Draynor Village Market, look for Fortunato, the wine seller. At first he denies having the vinegar, but after you mention that "he" sent you, he will allow you to buy vinegar from him. Get 8 vinegar.
Use the vinegar and bones on the pots then return to the Odd Old Man. Use a log on the pot boiler, then use a pot on it, and light it. Once it's done, remove the pot and repeat the process until every bone is polished.
Talk to the Odd Old Man to finish the quest!