The answer is complicated, but based on the ending of Dragon Ball GT, Goku's status is ambiguous; he's not explicitly dead, but transcends a normal mortal existence.
Here's a breakdown:
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The Apparent Demise: During the battle with Omega Shenron, Goku seemingly dies after a devastating attack. This occurs around episode 63 of Dragon Ball GT.
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The Resurrection and Transformation: After the battle, Goku isn't revived with the Dragon Balls. Instead, he merges with the Dragon Balls, essentially becoming a Dragon God or a guardian of the Dragon Balls. He transcends his mortal existence.
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Departure with Shenron: Goku is then seen riding away with Shenron, the Eternal Dragon, to restore the Dragon Balls to their original power. This is not a physical death in the traditional sense.
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Implications: He's not accessible in the same way he was before. He can appear and disappear at will, as shown when he attends the World Martial Arts Tournament decades later and then leaves again.
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Ambiguity: Whether this counts as "alive" is debatable. He's no longer simply a Saiyan warrior, but something more. He's moved beyond the mortal realm and taken on a different role. While not dead, he's also no longer "living" in the conventional sense.
In conclusion, while Goku appeared to die in the battle against Omega Shenron, he ultimately transcended death by merging with the Dragon Balls. He is no longer alive in the traditional sense, but exists as a protector and guardian connected to the Dragon Balls, effectively becoming a divine being.
