Bears? You mean bars, it's a metal that you usually mine with the Blacksmith to get a Bar, but in the Castle you can get a whole bar by doing a Treasure Map just like the ones from the Haven Museum.
Buy the Map, then click the icon for your Quest Log (!) but instead of clicking the ! inside that you click the treasure chest.

Or you could do like most people did and just mine those every 3 days 20 hours to get enough for the Robo Claw, it took me 3 weeks, but I had some extra Adamantium prior to that.

Anyone else reading this and not yet in the Castle, mine them as often as you can and you will be ready for the horrid requirement of 16 bars when you need them.

Try this website:
http://www.cafepress.com/+adamantium+teddy_bears