NB: I'm only going to be really speaking about this in an European context - I know tragically little about the history outside of that context.

Okay so it depends entirely on what you consider a museum to be, because then your answer could be "as old as recorded history" or "the 16th century" or "they started in 1753."

So let's start at the very beginning. Royal collections of things have been going on since, well, Royalty was a concept. You have memorials and inscriptions carved onto Babylonian reliefs commemorating victories, ascensions, transfers of power, births, deaths, etc. You have collections of statuary and gems and objects that are kept - privately - for the recollection of memories and safekeeping of loot (which becomes, let's not put too fine a point on it, the purpose of museums in the Anglosphere for about 250 years, once they start up and running)

These royal collections extend as far as royalty goes - they're private, they belong to the Crown, and they're there mostly for other royalty to gawk at, but they're there. Things are kept safe and maintained and repaired, mostly. Then in the early 16th century you start getting what are called kunstkamer, the Cabinet of Curiosities.

Kunstkammer, much like royal collections, are ways for the wealthy to show off neat and weird things they've found or been given or purchased. The major difference between a Royal Collection and a kunstkammer is how many people could see it at any one time. While the owners were wealthy (and still are - don't let anyone try and convince you that a private collection is in any way really different from a kunstkammer) and generally only invited other wealthy folk to see, the sheer size of viewing population that they're drawing from is much larger than those who would be permitted to view a royal collection. This, of course, helps spread the idea of a collection of weird, cool things across Europe, and that in turn starts to fire up the public imagination of what these things could be holding.

Then 1753 hits. And the British Museum is formed.

Now, the BM wasn't always open to the public in the same way it is now; you would need to get an invite to the museum, and those were relatively rare commodities, but if you could afford it, you could see Sir Hans Sloane's collection, not to mention the Cottonian Library (where you got your Beowulf manuscript, amongst others) and the Harlean Library (which has Ptolemy in it, as well as a wealth of liturgical material.) In grand total, somewhere north of 71,000 objects were included in the original British Museum; by 1757, Parliament had added the Royal Library to the collection, and you really hit the ground running in terms of the whole thing being a Universal Museum, and other countries start following suit.

Of course, this isn't to say other types of collections have gone the way of the dinosaur. Private collections and Country Houses, for example, carry on the tradition of the kunstkamer, being collections of weird things the owner (whether current or original) thought were worth showing. Specialized collections, too, carry on that tradition, whereas the BM's idea of a Universal collection (that is, a collection where you have things from all aspects of life - art, science, history, technology, anthropology, archaeology, etc.) is very very different, owing in part to scale and finance, and in part to the directives of the museum itself.

If you're looking for really good reads on the development of museums in general, hare are a couple of ones I highly recommend:

Bennet, Tony. The Birth of the Museum: History, Theory, Politics. Routledge, 1995.

Findlen, Paula. Possessing Nature: Museums, Collecting, and Scientific Culture in Early Modern Italy. University of California Press, 1996

Karp, I. & Kratz, C. (eds) Museum Frictions: Public Cultures/Global Transformations. Duke University Press, 2006