It’s a little hard to answer this, because ‘horror’ is a very generic term, and in particular has a lot of connotations derived principally from modern horror films. The Greeks certainly didn’t have jump-scares, at least not in their drama. In fact, they avoided any killing – no matter how horrifying – on stage at all: with the possible exception of Sophocles’ Ajax, every death that we know of in Greek tragedy happens off-stage, and is then reported to characters on stage by a messenger (and this motif is sufficiently common to be a very well-recognised trope of Greek tragedy, the messenger speech).
Nonetheless, if we take ‘horror’ to be some act that involves a specifically repulsive type of violence (so distinct from e.g. violently killing enemy soldiers in battle), some features we expect from modern horrors (whether written or performed) do appear in ancient Greek literature and drama. The Odyssey has a particularly famous example, when Odysseus and his men arrive at the cave of the Cyclops. There’s an ominous feeling building as the Greeks enter the empty cave and – breaking Greek guest customs – eat the Cyclops’ cheeses. When the Cyclops finally appears, Odysseus and his men are immediately afraid (Odyssey 9.236 δείσαντες), and retreat to the back of the cave, but this doesn’t save them for long. After speaking to Odysseus for a while, and making clear he will not spare his men, the Cyclops finally acts (9.288-93):
ἀλλ᾽ ὅ γ᾽ ἀναΐξας ἑτάροις ἐπὶ χεῖρας ἴαλλε,
σὺν δὲ δύω μάρψας ὥς τε σκύλακας ποτὶ γαίῃ
κόπτ᾽: ἐκ δ᾽ ἐγκέφαλος χαμάδις ῥέε, δεῦε δὲ γαῖαν.
τοὺς δὲ διὰ μελεϊστὶ ταμὼν ὡπλίσσατο δόρπον:
ἤσθιε δ᾽ ὥς τε λέων ὀρεσίτροφος, οὐδ᾽ ἀπέλειπεν,
ἔγκατά τε σάρκας τε καὶ ὀστέα μυελόεντα
He leapt up and laid his hands upon my companions, and having gathered up two of them, he dashed them on the ground like puppies; their brains flowed out to the floor, and the ground grew wet. He cut them up limb from limb and prepared his meal, and like a lion reared in the mountains, he left nothing, but ate the entrails and flesh and bones full of marrow.
The suddenness of the monster’s actions (actually perhaps quite close to a jump-scare), and the pathetic comparison of Odysseus’ soldiers to puppies, emphasise the overwhelming power of the Cyclops. The unnecessarily extensive detail – an entire line dedicated to specifying that every part of these men was eaten – makes it really horrific, and casts the Cyclops as a terrifying monster. In an oral recitation, it’s easy to see how scary this moment could be.
As noted, most of the truly horrific moments in Greek tragedy happened off-stage, whether that’s Medea having her husband’s new wife burn to death and murdering her own children (Euripides’ Medea), Heracles going mad and murdering his own family (Euripides’ Heracles), or Heracles being burned to death by centaur’s blood (Sophocles’ Trachinian Women). On-stage it’s much rarer, but a good example comes in Aeschylus’ Eumenides, the final play of his Oresteia trilogy. Orestes, having murdered his mother, is now pursued by the Furies (Eumenides), horrific monsters that torment those who have committed great crimes. At the beginning of the play they are roused by the ghost of Orestes’ mother Clytemnestra, who appears as a character herself – this is highly unusual for Greek tragedy. After pointing out the wounds on her chest that killed her, she finishes her exhortation to the Furies (lines 137-9):
σὺ δ᾽ αἱματηρὸν πνεῦμ᾽ ἐπουρίσασα τῷ,
ἀτμῷ κατισχναίνουσα, νηδύος πυρί,
ἕπου, μάραινε δευτέροις διώγμασιν.
You, send after him a gale of bloody breath, waste him away with the vapour, the fire from your womb, follow him, quench him with a second pursuit.
Again there’s a palpable horror to this scene: the explicit goriness combined with the eerily supernatural ghost and divinities make this highly repulsive and terrifying. It is another good example of horror being used in Greek literature.
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