What exactly was a "mathom house" to a hobbit.

[Melilot Brandybuck]

I found it interesting that Bilbo stored his mithril chain mail that he acquired in his journeys, in a mathom house. From what I understand a mathom house is like a museum, but in our society a museum is somewhere precious artefacts from the past are displayed. As the hobbit society scoffed at Bilbo's tales, and were generally uncomfortable with the fact that he was well-travelled, I wonder what they thought of it being displayed in the mathom house, and did they consider it an item of significance. Did they look at it as the work of art it truly was? What compelled Bilbo to loan it to the mathom house, knowing the way the other hobbits viewed his travels? Also, considering the value of the item, why wasn't it stolen? Wouldn't word have got around that such a precious object was there? It leads me to wonder what was a mathom house, to a hobbit?