Rome is far off. You are not Rome
So the problem with me and lists is that I want to make them comprehensive. Therefore this is not a catalogue of all the fiction with classical settings that exists, or even all the fiction with classical settings I've read or own. It is an attempt at listing some of my favorites of the genre without falling down the sinkhole of completism. If you don't see your favorites, please feel free to name them in comments. It may be that I've accidentally left them out. It may be that my local library does not contain enough Naomi Mitchison. It may also be the case that I'm not sure I need to re-read all of The Bull from the Sea ever again. This endeavor brought on by recent discussion of Dorothy J. Heydt's Cynthia stories.
Genres considered for inclusion are historical fiction and historical fantasy, with a flexible exception for contemporary narratives in which the classical past plays a prominent part. I have tried to stay away from mythological retellings because otherwise we'll be here all day. Honorable mention at the end goes to alternate histories or secondary worlds drawn so strongly from classical history that they feel like it.
To explain the minimal organization in the list below: novels or stories in the same series are listed all on the same line; independent novels or stories by the same author are given their own carriage return; I've put links to stories online where they exist. There are almost certainly missing entries I'll wish I'd listed in the morning. I'm sleeping about three hours a night these days.
Julia August, "Elephants and Omnibuses" (Lackington's #2, 2014)
Robin W. Bailey, "Child of Orcus" (Sword and Sorceress, 1984)
Samuel R. Delany, Phallos (2004)
Gemma Files, "Sent Down" (The Worm in Every Heart, 2004)
Gemma Files, "Villa Locusta" (The Harrow 10.1, 2007)
Alan Garner, Red Shift (1973)
Théophile Gautier, "Arria Marcella: Souvenir de Pompéi" (La Revue de Paris, 1852; translated by Richard Holmes as "The Tourist" in My Fantoms, 1976)
Robert Graves, I, Claudius (1934)
H.D., Palimpsest (1926)
Dorothy J. Heydt, "Things Come in Threes" (Sword and Sorceress, 1984) and all succeeding Cynthia stories (Sword and Sorceress III—IV, VI, IX—X, XIII—XVII, XIX—XXI, 1986–2004)
Tom Holt, Goatsong (1989) and The Walled Orchard (1990)
Rudyard Kipling, Puck of Pook's Hill (1906)
Karen K. Kobylarz, "Cleopatra's Needle" (Paradox #5, 2004)
Ursula K. Le Guin, Lavinia (2008)
Tanith Lee, The Book of the Damned (1988) and The Book of the Beast (1988)
Tanith Lee, "Into Gold" (Isaac Asimov's Science Fiction Magazine 10.3, 1986; collected in Woman As Demons, 1989)
Tanith Lee, "Sirriamnis" (Unsilent Night, 1981; reprinted in The Gorgon and Other Beastly Tales, 1985)
Shweta Narayan, "Eyes of Carven Emerald" (Clockwork Phoenix 3, 2011)
John Maddox Roberts, SPQR (1990), The Sacrilege (1992), The Temple of the Muses (1992)
Mary Renault, The King Must Die (1958)
Mary Renault, The Last of the Wine (1956)
Mary Renault, The Mask of Apollo (1966)
Mary Renault, The Praise Singer (1978)
Steven Saylor, Roman Blood (1991), The Venus Throw (1995), The House of the Vestals (1997), A Gladiator Dies Only Once (2005)
Rosemary Sutcliff, The Eagle of the Ninth (1954)
Thomas Burnett Swann, Where Is the Bird of Fire? (1970)
Harry Turtledove, "Goddess for a Day" (Chicks in Chainmail, 1995)
Jill Paton Walsh, Farewell, Great King (1972)
Evangeline Walton, She Walks in Darkness (2013)
As for the novels not appearing in our history—
Avram Davidson, The Phoenix and the Mirror (1969)
Guy Gavriel Kay, Sailing to Sarantium (1998) and Lord of Emperors (2000)
Tanith Lee, Mortal Suns (2003)
Melissa Scott, A Choice of Destinies (1986)
Megan Whalen Turner, The Thief (1996), The Queen of Attolia (2000), The King of Attolia (2006), A Conspiracy of Kings (2010)
The other problem with putting together this list is that it made me realize I want to edit an anthology of short fiction set in the classical world—I know the reprints I would want to ask for; I would want to hold a reading period for new material—and even if I knew a publisher to pitch it to, this is not a project for which I have time right now.
And I still don't really have any stories where the Carthaginians come off positively. Cato the Censor and the Aeneid cast a long shadow. Does anyone know any?
Genres considered for inclusion are historical fiction and historical fantasy, with a flexible exception for contemporary narratives in which the classical past plays a prominent part. I have tried to stay away from mythological retellings because otherwise we'll be here all day. Honorable mention at the end goes to alternate histories or secondary worlds drawn so strongly from classical history that they feel like it.
To explain the minimal organization in the list below: novels or stories in the same series are listed all on the same line; independent novels or stories by the same author are given their own carriage return; I've put links to stories online where they exist. There are almost certainly missing entries I'll wish I'd listed in the morning. I'm sleeping about three hours a night these days.
Julia August, "Elephants and Omnibuses" (Lackington's #2, 2014)
Robin W. Bailey, "Child of Orcus" (Sword and Sorceress, 1984)
Samuel R. Delany, Phallos (2004)
Gemma Files, "Sent Down" (The Worm in Every Heart, 2004)
Gemma Files, "Villa Locusta" (The Harrow 10.1, 2007)
Alan Garner, Red Shift (1973)
Théophile Gautier, "Arria Marcella: Souvenir de Pompéi" (La Revue de Paris, 1852; translated by Richard Holmes as "The Tourist" in My Fantoms, 1976)
Robert Graves, I, Claudius (1934)
H.D., Palimpsest (1926)
Dorothy J. Heydt, "Things Come in Threes" (Sword and Sorceress, 1984) and all succeeding Cynthia stories (Sword and Sorceress III—IV, VI, IX—X, XIII—XVII, XIX—XXI, 1986–2004)
Tom Holt, Goatsong (1989) and The Walled Orchard (1990)
Rudyard Kipling, Puck of Pook's Hill (1906)
Karen K. Kobylarz, "Cleopatra's Needle" (Paradox #5, 2004)
Ursula K. Le Guin, Lavinia (2008)
Tanith Lee, The Book of the Damned (1988) and The Book of the Beast (1988)
Tanith Lee, "Into Gold" (Isaac Asimov's Science Fiction Magazine 10.3, 1986; collected in Woman As Demons, 1989)
Tanith Lee, "Sirriamnis" (Unsilent Night, 1981; reprinted in The Gorgon and Other Beastly Tales, 1985)
Shweta Narayan, "Eyes of Carven Emerald" (Clockwork Phoenix 3, 2011)
John Maddox Roberts, SPQR (1990), The Sacrilege (1992), The Temple of the Muses (1992)
Mary Renault, The King Must Die (1958)
Mary Renault, The Last of the Wine (1956)
Mary Renault, The Mask of Apollo (1966)
Mary Renault, The Praise Singer (1978)
Steven Saylor, Roman Blood (1991), The Venus Throw (1995), The House of the Vestals (1997), A Gladiator Dies Only Once (2005)
Rosemary Sutcliff, The Eagle of the Ninth (1954)
Thomas Burnett Swann, Where Is the Bird of Fire? (1970)
Harry Turtledove, "Goddess for a Day" (Chicks in Chainmail, 1995)
Jill Paton Walsh, Farewell, Great King (1972)
Evangeline Walton, She Walks in Darkness (2013)
As for the novels not appearing in our history—
Avram Davidson, The Phoenix and the Mirror (1969)
Guy Gavriel Kay, Sailing to Sarantium (1998) and Lord of Emperors (2000)
Tanith Lee, Mortal Suns (2003)
Melissa Scott, A Choice of Destinies (1986)
Megan Whalen Turner, The Thief (1996), The Queen of Attolia (2000), The King of Attolia (2006), A Conspiracy of Kings (2010)
The other problem with putting together this list is that it made me realize I want to edit an anthology of short fiction set in the classical world—I know the reprints I would want to ask for; I would want to hold a reading period for new material—and even if I knew a publisher to pitch it to, this is not a project for which I have time right now.
And I still don't really have any stories where the Carthaginians come off positively. Cato the Censor and the Aeneid cast a long shadow. Does anyone know any?

no subject
Which is why I agree with you about the lack of good English translations, with the exception of Fagles because I haven't read that one yet. (I used to get into arguments with people in my department about Fitzgerald because they liked him for being literal and I thought it was a bad choice as a translation, which is illustrative.) I think Fagles is better at sounding literary and comprehensible as far as the Homeric epics go, though I only read the Iliad in the original. What I should really probably do is read the Fagles version with LeGuin's interpretation in mind, because I agree with you that she clearly gets Vergil really well.
no subject
Nice! What were the other empires?
(I used to get into arguments with people in my department about Fitzgerald because they liked him for being literal and I thought it was a bad choice as a translation, which is illustrative.)
. . . Fitzgerald is not literal. You can't use him as any kind of guide to the original grammar. You can't even sometimes use him as a guide to the original image; he elaborates. His lines always run over.
I think Fagles is better at sounding literary and comprehensible as far as the Homeric epics go, though I only read the Iliad in the original.
Usually I tell people to read Lattimore and Fitzgerald together: Lattimore is faithful and archaic, Fitzgerald is vivid and imagistic; if you read them by turns, you can approximate the dimensions of Homeric Greek. I've bounced off every one of Fagles' translations that I've tried, which I am faintly sorry about, but neither his Iliad nor his Odyssey nor his Aeneid works for me. It may be that I don't like his English, which just puts the kibosh on everything.
What I should really probably do is read the Fagles version with LeGuin's interpretation in mind, because I agree with you that she clearly gets Vergil really well.
I like that plan. Let me know how it works out!