(Telemachos:) 'Antinoös, I cannot thrust the mother who bore me, who raised me, out of the house against her will. My father, alive or dead, is elsewhere in the world. It will be hard to pay back Ikarios, if willingly I dismiss my mother. I will suffer some evil from her father, and the spirit will give me more yet, for my mother will call down her furies upon me as she goes out of the house, and I shall have the people's resentment.' (2.130-137)
The Furies are goddesses of vengeance and retribution, which is subtly—but importantly—different from justice. They're particularly invested in crimes against family, so Telemachos would seriously tick them off by kicking his mom out of the house.
(Telemachos:) '[…] fear also the gods' anger, lest they, astonished by evil actions, turn against you. I supplicate you, by Zeus the Olympian and by Themis who breaks up the assemblies of men and calls them in session: let be, my friends, and leave me alone with my bitter sorrow to waste away; unless my noble father Odysseus at some time in anger did evil to the strong-greaved Achaians, for which angry with me in revenge you do me evil in setting these on me.' (2.66-74)
Themis is the Greek goddess of something like social order—the way things are done, good conduct, divine law. By invoking Themis, Telemachos is reminding the suitors that what they're doing isn't just super annoying and insulting to him—it's an offense against the gods.
(Polyphemos, in Odysseus' tale:) '"Hear me, Poseidon, who circle the earth, dark-haired. If truly I am your son, and you acknowledge yourself as my father, grant that Odysseus, sacker of cities, son of Laertes, who makes his home in Ithaka, may never reach that home; but if it is decided that he shall see his own people, and come home to his strong-founded house and to his own country, let him come late, in bad case, with the loss of all his companions, in someone else's ship, and find troubles in his household." 'So he spoke in prayer, and the dark-haired god heard him.' (9.528-536)
Polyphemos wants revenge. But is it justice? Zeus lets it happen—for a while, at least—so we're inclined to think that maybe it is. The question is whether it's Odysseus' punishment for blinding the guy, or for being dumb enough to reveal his name.
(Helios, in Odysseus' tale:) '"Father Zeus, and you other everlasting and blessed gods, punish the companions of Odysseus, son of Laertes; for they outrageously killed my cattle, in whom I always delighted, on my way up into the starry heaven, or when I turned back again from heaven toward earth. Unless these are made to give me just recompense for my cattle, I will go down to Hades' and give my light to dead men."' (12.377-383)
Okay, notice Helios' words "just recompense." We're used to a system of justice where "recompense" means either a fine or a sometime behind bars. But Helios is working from a more eye-for-an-eye sense of justice: they killed my cattle, and now I want them dead.
(Athene:) 'It is true that the young men with their black ship are lying in wait for him to kill him before he reaches his country; but I think this will not happen, but that sooner the earth will cover some one of those suitors, who now are eating away your substance.' (13.425-428)
Athene's "I think" is more like, "I'll make sure." She is the goddess of justice, after all. If she says that something is right, it's right.
(Alkinoös:) ‘[…] let us man by man each one of us give a great tripod and a caldron, and we will make it good to us by a collection among the people. It is hard for a single man to be generous.’ (13.13-15)
Alkinoös believes that repaying Odysseus for all that he has lost – his men, twenty years of his life – is only justice for the poor man’s suffering.
[Antinoös] was to be the first to get a taste of the arrow from the hands of blameless Odysseus, to whom he now paid attention as he sat in Odysseus' halls and encouraged all his companions. (21.98-100)
Check out this "was to be the first." The passive construction here makes it seem like Antinoös' death is due to some sort of divine justice that neither men nor gods can control. But is that really true? Athene's hand is all over this.
(Odysseus:) 'Keep your joy in your heart, old dame; stop, do not raise up the cry. It is not piety to glory so over slain men. These were destroyed by the doom of the gods and their own hard actions […].' (22.411-413)
You have to draw the line somewhere. It's just to mercilessly slaughter the suitors, but it's not just to gloat about it. The fact that it's just is actually what makes gloating wrong.
(Odysseus:) 'O son of Polytherses, lover of mockery, never speak loud and all at random in your recklessness. Rather leave all speech to the gods, since they are far stronger than you are. Here is your guest gift, in exchange for that hoof you formerly gave to godlike Odysseus, as he went about through the palace.' (22.287-291)
As the herdsman Polytherses kills nasty suitor Ktesippos, he makes sure the guy knows exactly why this is happening: it's "in exchange" for the time he threw the hoof at Odysseus. It's another example of cause-and-effect justice.
(Odysseus:) 'You dogs, you never thought I would any more come back from the land of Troy, and because of that you despoiled my household, and forcibly took my serving women to sleep beside you, and sought to win my wife while I was still alive, fearing neither the immortal gods who hold the wide heaven, nor any resentment sprung from men to be yours in the future. Now upon you all the terms of destruction are fastened.' […] [A]ll that you have now, and what you could add from elsewhere, even so, I would not stay my hands from the slaughter, until I had taken revenge for all the suitors' transgression. Now the choice has been set before you, either to fight me or run, if any of you can escape death and its spirits. But I think not one man will escape from sheer destruction.' (22.35-41, 62-67)
Here's your classic "if-then" causality: if you despoil a man's house, rape his servants, and come on to his wife, then you will be slaughtered. But is that the way justice works? Should work?
(Athene:) 'Hold back, men of Ithaka, from the wearisome fighting, so that most soon, and without blood, you can settle everything.' (24.531-532)
Oh, sure. Now that Athene's got what she wants, she's ready to stop fighting. When her precious Odysseus is the one being threatened, she suggests that maybe they should stop taking revenge on each other and start thinking about new ways to find justice.