The modern novel is a relatively recent concept, so while there are novels from antiquity, they are somewhat different to what most people are used to. However since you use the epic poem of the Iliad as an example, a text which is definitely not the easiest to master, I guess that is not going to be a problem.

We have a handful of novels from the Hellenistic times, and although most of them are very standard formulaic and idealised romances of the Greek variety, there are a couple of exceptions.

My personal favourites are The Golden Ass by Apuleius and Satyricon (sadly only fragments are extant) by Petronius. Most surviving texts from the period was written by an elite for an elite, and thus rarely offers any insight into the goings on of the less fortunate persons in Greco-Roman culture. And that is exactly where these texts stands out. We get glimpses of the lives of vagabond travellers, sleazy inns and the seedier parts of Roman towns. For the same reason I also enjoy Juvenals third Satire, depicting the poor and scorned poet living on the top floor apartment (where the poor usually lived) in Rome, complaining about immigrants and his lack of success. Again a rare glimpse into a world that did not leave many written sources behind.

In a completely different vein, but still extremely interesting, we have the True History of Lucian of Samosata, which is basically a fantastic story of a man travelling to the moon, witnessing the wars of the people living there and further voyages to imaginary worlds. Lucian was an extremely interesting writer, always coming up with some really imaginery stuff, and while playing on older literary traditions, he managed to create his own particular style which still holds up to this day with respects to entertainment and readability.

For the traditional greek romance novel, all basically cast in the same formulaic mold, I mostly recommend the Aethiopica by Heliodorus of Emesa. It contains a wonderful captivating introductory chapter, which unfortunately promises more than the rest of the book can live up to, but it is still a good yarn about the struggles of two lovers continously being interrupted by unfortunate events which they have to overcome.

Edit: Regarding secondary literature I can highly recommend Thomas Hägg - The Novel in Antiquity (1991). Hägg was pioneer in the study of this subject, and while his work has inspired a lot of subsequent studies and discoveries, his book remains a very good introduction to the ancient novel, not to mention a very good read in itself if you are into that sort of stuff.

So /u/breecher has given you a fabulous write up on the differences between novels and 'ancient literature', as well as some recommendations. I'm just adding my own figurative two cents here, but I am nothing close to an expert.

Plato's Republic - A substantive exploration into philosophy and justice. Examinations of why we abide by laws, and if we are truly happy by following the laws, and how we really just define the laws we follow. It's less of "fiction" but rather an extended dialogue (as it is defined as a Socratic dialogue) - a debate or conversation between various people.

Ovid's Metamorphoses - poems which have inspired names that you may be more familiar with, such as Dante, Chaucer, or Shakespeare.

Sophocles' Oedipus Rex - spawning the more famous term 'oedipus complex' as coined by Sigmund Freud. A work that is, in some ways, the benchmark tragedy that all follow - that man, despite having everything, will still lead himself, or be lead, to his downfall. You see common plot themes or elements in the tragedies of many other authors that followed.

Sophocles' Antigone - Great drama with well-developed characters, and yet Sophocles manages to examine themes that remain significant in our contemporary context, such as going against the law or what is "right" to act upon personal interest, or to reveal and warn of man's hubris.

If you are interested in something even older than the Hellenistic and Roman era works listed in the other answers, I suggest the Epic of Gilgamesh. It's a legend, a myth, but because the person of Gilgamesh is purported to have existed in one form or another (he appears in the King Lists), and the story is very much a fantastical account of his exploits, I suggest that it is one of the earliest works of literature. It has a number of very interesting themes, as well, such as immortality, friendship, and civilization. There are a number of great modern translations of the story as well.

On the Egyptian side of things, there is the Story of Sinuhe, a work that was very popular in its own time (~20th century BC, 12th Dynasty of Egypt). It, too has a number of literary themes such as divine providence and mercy. The anonymous author has been called the "Egyptian Shakespeare", if that gives you any indication of the nature of the work.

The Chinese adored writing stuff, but novels were a more recent development (mainly during the Ming dynasty. Nonetheless here are some literary favourites passed down from long ago.

The Zuo Commentary 左传 is the earliest documented prose recording of history, expanded from the Spring Autumn Annals 春秋 of the eponymous Spring Autumn period. Since it's a commentary, it necessarily adds a little interpretation to history, with slight embellishment here and there. Another popular read was the History Classic 书经, a collection of speeches by the leaders of the ancient Chinese periods of Yu, Xia, Shang and Zhou. Although the actual date of origin is still debated, it should be written no later than 4BC.

Records of the Grand Historian 史记 is probably the most famous Chinese history book, but it also started the trend of writing biographies of important people 列传 using anecdotes, which pretty much reads like fables. There were also compilations of folk tales like Investigations of the Supernatural 搜神记 compiled in the 4th century, although nothing of novel length yet.

People in ancient China used written words more for ideological matters, with plenty of philosophical treatises and military texts written. Analects 论语 by Confucius and Moral Classic 道德经 are great examples of the former, Six Secret Teachings 六韬 and the Art of War 孙子兵法 are classics of the latter.

The most similar to Homer's works would be the Tang and Song dynasty poems. It was only then when the Chinese started the widespread use of the written word for recording stories. Among those, my personal favorite is Song of Everlasting Regret 长恨歌 by Bai Juyi, detailing the tragic tale of concubine Yang 杨贵妃 of the Tang dynasty.

If you wish for actual prose novels, than the Ming dynasty is the favorite. Three of the four classics were written then (the Romance of the Three Kingdoms 三国演义 a historical novel much like Illiad in prose form, Water Margin 水浒传 Robin Hood meets Braveheart of sorts, and Journey to the West 西游记 a supernatural tale on the quest to meet Buddha, with *Dream of the Red Chamber * 红楼梦 the political struggles in a woman dominated family written later). Other popular works include Cases of Judge Bao 包公案 ancient China's Law and Order, the Plum in the Golden Vase 金瓶梅 a graphic asexual novel often banned, and Investiture of the Gods 封神演义 mythological take on the fall of Shang and rise of Zhou.

Translations are available online of most titles I listed here, though some lose the original flavor in English, especially if they are not written in prose.