Before the British Museum
... there were private collections.
From John Evelyn's diary, December 16, 1686.
"I carried the Countesse of Sunderland to see the rarities of one Mr Charleton in the Middle Temple, who shew'd us such a collection as I had never seene in all my travels abroad, either of private gentlemen or princes. It consisted of miniatures, drawings, shells, insects, medailes, natural things, animals (of which divers, I think 100, were kept in glasses of spirits of wine), minerals, precious stones, vessells, curiosities in amber, christal, achat, &c; all being very perfect and rare in their kind, especially his bookes of birds, fish, flowers and shells, drawn and miniatur'd to the life. ... This gentleman's whole collection, gather'd by himselfe travelling over most parts of Europe, is estimated at £8000."
A note in my edition says this collection was purchased by Sir Hans Sloane and hence became part of the foundation collection of the British Museum.
I was particularly taken by this since I work (voluntarily) every fortnight in the BM's new Enlightenment Gallery, which recalls this early comprehensive approach to collecting. The juxtaposition of times, cultures and modes of thought that it creates can indeed be enlightening.
I do handling, which means having ancient objects that visitors can hold -- our oldest human-made one is a 350,000-year-old handaxe. Holding it really does make history come alive.