Arachnophobia combines arachno- and -phobia. The deeper Greek words are arachnē, spider or spiderweb, and phobos, fear.
Phobos also meant flight in the sense of fleeing; its related Greek verb phebesthai meant to flee. The modern fear word uses these older components.
Sources
- Merriam-Webster: arachnophobia
Word History explicitly gives arachno- plus -phobia.
- Merriam-Webster: arachno-
The combining-form entry gives Greek arachne, spider or spiderweb, and the arachnology example.
- Merriam-Webster: -phobia
The noun combining-form history reaches Greek phobos, fear, through Greek -phobos and Latin formations.
- Online Etymology Dictionary: -phobia
Also built on -phobia, from Greek *phobos* "fear": agoraphobia, acrophobia
- Online Etymology Dictionary · arachnophobia
Meanings of the older forms on the word path.