I would also look at the Greek playrights Euripedes, Sophocles and Aristophanes. Yes it is easier to see them performed than to read them, but new translations are available all the time.
These writers have very modern sensibilities: A few years ago I saw the Abbey Theatre Group do Euripides’ Medea , with Fiona Shaw in the title role. This is a truly modern story of “a woman scorned.” After the play Ms. Shaw hung around on stage and answered questions about the great Greek writers. It was her opinion that it wasn’t until the Elizabethean/Jacobean writers, (Shakesphere, Marlowe, Bacon, etc), that these writers are equaled.
The two great Indian epics, Ramayana and Mahabharata were written (or compiled) around the beginning of the common era. Both are highly entertaining and very popular even today. The Mahabharata is very long, though; something like 90,000 couplets IIRC. That is significantly longer that the Illiad and the Odyssey put together. A tough read, but worth it. The Ramayana is shorter - around 24,000 couplets - and a lot more “manageable”. The story is more coherent, and it is highly entertaining. Both works are widely available todai - I think you can find Ramayana in a Penguin Paperback edition.
There is also a very old Tamil epic called The tale of an anklet or Silapattikaram from around the 4-5th century CE. It has a nice story and is very informative if you are interested in South Indian history. Also it is a lot shorter than the two mentioned above.
And I second Gilgamesh - a wonderful, timeless work. Simply amazing.
Many works have been called the oldest novel. Don Quixote is often given, but Gargantua is in essentially the same form and was published earlier. Similarly, The Tale of Genji gets mentioned but it owes a lot to the popularity of the earlier Tale of the Bamboo Cutter. All of those are much more recent than Roman works like Satyricon, which I nominate.
I also found Plato to be quite legible, certainly more so than the majority of more recent philosophers.
I’m partial to Virgil’s Aeneid, but then I read the first half in the original Latin. The most popular translation obviously doesn’t show off all of the poetry of the original, but it’s poetic in it’s own right.
Some of Plato’s dialogues are surprisingly good reading, at least if you have a good translation that lets the liveliness and sarcasm come through.
The Kama Sutra is from at least the 2nd century, and I suspect that people are still reading it.
To be honest some parts are very entertaining.