Paul Oskar Kristeller famously argued in "The Modern System of the Arts" (Problems in Aesthetics ed. Morris Weitz, The Macmillan Co., 1970, 108-164) that the "system of the five major arts, which underlies all modern aesthetics and is so familiar to us all, is of comparatively recent origin and did not assume definite shape before the eighteenth century, although it has many ingredients which go back to the classical, medieval and Renaissance thought." (110) This has largely been accepted as proven. However it should at least be modified since Plato's Ion pretty much provides the basis for the system. The five major arts are, according to Kristeller, "painting, sculpture, architecture, music and poetry." Plato assumes most of this system in the Ion, although in the Jowett translation, the word "art" translates "techne" and is reserved for thing that we would not consider to be part of the system of major arts, whatever that list might be. For example, techne includes medicine and charioteering. I want to make one caveat here: the specific argument of the Ion is that the poet and the rhapsode do not operate from art but from inspiration, and others of what we would call fine arts are brought up, but in contrast to these two. However, it is clear that Plato is making his own point here and, in doing so, is pretty much assuming a system of fine arts as a starting point. That he does not use a term that can be translated as "fine art" does not prove Kristeller's point. The bottom line is that most of the group Kristeller mentioned as fine arts are clumped together in the Ion.
Socrates starts the line of argument by saying "poetry is a whole" and then moving from there to the other arts which also constitute the whole. The point is that Ion should be able to judge and interpret other poets than Homer since poetry is a whole. We only learn later that he can't do this because his ability, at least hypothetically, is based on inspiration rather than rule-following skill or knowledge. However, many interpreters of the Ion have, I think justly, seen the inspiration theory as ironic and have insisted that Socrates is not complimenting Ion by saying that he is out of his mind when he practices his craft.
Socrates then says "when any one acquires any other art as a whole, the same may be said of them." (47) So, after poetry, the next art discussed is painting, followed by sculpture, flute-playing, harp-playing, and the art of the rhapsode (these last three in one paragraph, so grouped together) . In each case he mentions specific individuals famous in these art forms, for example Polygnotus in painting, Daedalus in sculpture, Thamyris a mythological lyre-player, Orpheus another mythological lyre-player, and Ion himself, in the art of the rhapsode. So how many of the fine arts do we have so far? It seems that we have painting, sculpture, music and poetry, although the category slicing is somewhat different from our own when it comes to music and poetry, two art forms that the Attic Greeks found difficult to think apart from each other. Poetry is definitely included since the rhapsode is included, and the art of the rhapsode is closely associated with the art of poetry in the dialogue. Moreover, the art of mimetic poetry is also closely associated with the art of painting in the Republic. The only of the classic five excluded from Plato's list is architecture. Plato does talk about architecture in The Statesman where he describes it as a science, although distinct from calculation. It is a science in which the architect's business is to insure that his workers follow his instructions to completion.
Kristeller says "if we want to find in classical philosophy a link between poetry, music and the fine arts, it is provided primarily by the concept of imitation" (115) which is true, but the point should be addressed cautiously. Mimesis is not actually mentioned in the Ion passage discussed, and Plato only includes comedy and tragedy, i.e. theater in general, under the mimetic arts in the Republic. Kristeller mentions this in a footnote (115) when he observes that only certain kinds of poetry are imitative for Plato, but note again that Plato does talk about non-imitative poets, for example Archilochus. He even discusses three kinds of poetry in the Republic in Book III when he discusses style: imitative poetry (comedy and tragedy), non-imitative poetry (dithyramb) and mixed (epic).
Kristeller's argument seems to rest on the idea that architecture is not included. But this is not convincing since the list of fine arts has changed more than once over history, with architecture being added later along with dance and gardening, and with gardening dropping out even later. Today, the concept of "fine art" is seen as somewhat anachronistic and to the extent that there is any canonical group it would include any aspect or part of any discipline that seems sufficiently like what has previously been called fine art: so this would include some gardens, Japanese tea ceremonies, some video games, and so forth.
Kristeller also mentions that music and dance "are treated as parts of poetry and not as separate arts" which is true, but again no real argument for his overall claim. Also it is not as though Plato does not say specific things about dance and music distinct from claims made about poetry more generally. More shockingly, Kristeller seems to think that Plato would group the imitative arts with the use of the mirror and with magic tricks, failing to see that this is not a matter of categorization but rather a rhetorical move on Plato's part. (116) In the Ion when he talks about arts as a whole he does not include these categories: there are no famous users of mirrors or magic tricks in the way that there are famous poets etc.
Kristeller's confusion about Plato is further evidenced in a footnote in which he says that Plato "arrived at his distinction between productive and imitative arts without any exclusive concern for the 'fine arts' since imitation is for him a basic metaphysical concept which he uses to describe the relation between things and Ideas." (115) This is wrong: Plato clearly distinguishes the relation between the bed made by the carpenter and the Idea of the bed from that between the painting of the bed and the physical bed (in Book X of the Republic). He only calls the second "imitation" strictly speaking.
So, based on Plato's writings at least, there did seem to be an implicit list of the fine arts in ancient Greece, one not based on Plato's notion of mimetic art or even on some notion of arts of inspiration.