What does Eurylochus mean when he says, "I’d rather die at sea, with one deep gulp of death, / than die by inches on this desolate island here!"
The answer is that Eurylochus was attempting to convince the rest of Odysseus' men to slaughter and eat the sacred cattle of Helios, the sun-god. Circe had warned Odysseus that his ships would be destroyed if he and his men ate Helios' cattle. Consequently, Odysseus made his men swear to leave the cattle alone.
However, the ship's stores began to run low. The men began to starve. Eurylochus—Odysseus' second-in-command—mutinies and attempts to convince the men to eat the cattle, even though they know that it might lead to their deaths. He says:
Listen to me, my comrades, brothers in hardship. All ways of dying are hateful to us poor mortals, true, but to die of hunger, starve to death—that’s the worst of all. So up with you now, let’s drive off the pick of Helios’ sleek herds, slaughter them to the gods who rule the skies up there.
If we ever make it home to Ithaca, native ground, erect at once a glorious temple to the Sungod, line the walls with hoards of dazzling gifts! But if the Sun, inflamed for his longhorn cattle, means to wreck our ship and the other gods pitch in—I’d rather die at sea, with one deep gulp of death, than die by inches on this desolate island here!
Eurylochus knows he is risking death by eating the cattle. However, he believes he is facing starvation (even though Circe had promised they would return safely if they left the cattle alone). After comparing a quick death—drowning—with a prolonged death—starvation—Eurylochus says that he would rather die quickly—"with one deep gulp of death"—than slowly—"by inches."