Early poets like Homer probably didn't archive their poetry at all: some scholars think they did, but that's largely because of a refusal to admit the reality of oral transmission.

We have no testimony about books in written form earlier than the second half of the 500s BCE.

Epigraphic evidence tends to suggest that's also when there was a phase-transition in the function of writing -- surviving inscriptions show a transition in the 'voice' they adopt after 550-540, from words designed to be uttered at the moment they're read, towards words as an archive of something said in that past. There's an interesting discussion of this in chapter 1 of Jesper Svenbro's 1993 book Phrasikleia. An anthropology of reading in ancient Greece (orig. in French, 1988).

I wrote an answer last month that talked about what Homer looked like prior to the 400s BCE, or rather what modern scholars have speculated Homer looked like.

After that time, scrolls were absolutely the standard medium for all books, until the codex form (pages bound at a spine) began to take hold in the 2nd-4th centuries CE. A single scroll containing the entirety of something like Herodotos' or Thucydides' histories would indeed be too unwieldy: that appears to be the reason why things like that are divided into 'books' -- 9 'books' in Herodotos' case, 8 for Thucydides. These work out to

Author	No. of books	Characters (Greek)	Words (Greek)	Characters per book	Words per book
Thucydides	8	796,823	150,324	99,603	18,791
Herodotos	9	963,954	185,637	107,095	20,626

('Characters' means only letters of the alphabet: ancient books didn't normally have word-spaces or punctuation.)

Now, that's prose books. The figures for poetry look quite different --

Author	No. of 'books'	Characters (Greek)	Words (Greek)	Characters per 'book'	Words per 'book'
Homer	48	979,663	199,260	20,410	4151
Apollonios	4	207,186	39,086	51,797	9772

So I'd say it's doubtful whether they were split up into a separate scroll for each 'book'. But that's something where we don't have clear evidence.

For reference, I obtained these figures by opening up the relevant files in the TLG-E database and using regular expressions to remove most non-alphabetic characters for the word count, then the spaces too for the character count. I notice the Chicago Homer website gives a word-count for Homer of 198,803 words: the TLG figure is 0.23% higher. I'm not going to spend the time to figure out why, I'm sorry!

Edit: I should have mentioned, there are rare cases where we know that a book was divided across more than one scroll: books 1 and 17 of Diodoros of Sicily, in particular. It's hard to say how common that would really have been.

Edit 2: Just as a potentially interesting titbit of further information: if we suppose that 100,000 characters per scroll (prose) represents a normal figure, that would imply that the library of Alexandria in Kallimachos' time, which he reported to have 532,800 scrolls, would represent 49.6 GiB in terms of modern digital storage.