It entirely depends on the style and aesthetics of the translator. Any translation of anything can be made to sound vivid and modern, or to sound stilted and archaic. As an example, here are five translations of Iliad 9.9-13. Lombardo (1997):

Agamemnon's heart was bruised with pain
As he went around to the clear-toned criers
Ordering them to call each man to assembly,
But not to shout. He pitched in himself.
It was a dispirited assembly.

Lang, Leaf, and Myers (1897):

But Atreides was stricken to the heart with sore grief, and went about bidding the clear-voiced heralds summon every man by name to the assembly, but not to shout aloud; and himself he toiled amid the foremost. So they sat sorrowful in assembly...

Pope (1720):

Forth stalk'd Atrides with heart-riving wo
Transfixt; he bade his heralds call by name
Each Chief to council, but without the sound
Of proclamation; and that task himself
Among the foremost sedulous perform'd.
The sad assembly sat;...

Butler (1898):

The son of Atreus in dismay bade the heralds call the people to a council man by man, but not to cry the matter aloud; he made haste also himself to call them, and they sat sorry at heart in their assembly.

Johnston (2006):

Atreus' son, heart overwhelmed with painful sorrow,
went to give out orders for clear-voiced heralds
to summon all the warriors to assembly,
calling them one by one, not with a general shout.
He himself, with his heralds, carried out the task.
The counselors sat heart sick.

You can do this with any text in any language, regardless of place, people, or period. Style is always the choice of the translator. Sometimes a translator will try to render the style of the original as s/he perceives it, but that is still a choice.

No one ever spoke the language and style of the Iliad in everyday speech. If you were to try to give an accurate rendering of its style in modern English, it would sound something like an uptown version of rap, accompanied by a standard concert instrument (a guitar, say), and in a mix of dialects (say, New Jersey, Los Angeles, and Tennessee).

There are a few possible answers depending on the text. Two things I have direct knowledge of:

• Some authors try to keep the original meter, or some form of it. We're not really used to this in English. It gets harder if they try to keep the prosaic language we expect from long stories. This is hard enough without it being a translation to begin with.

• Especially in the case of Homer, the text would have sounded archaic in antiquity. The Greek of Homer was not the Greek of Thucydides. Some translators try to replicate this, choosing an archaic English word where an archaic Greek one was used.