Relevant quotes from the serial are below. For those unaware, the Great One is a talking telekinetic spider the size of your house.
GREAT ONE [OC]: You are proud, little man. I see that I shall have to teach you to have respect. Round you go, Doctor.
DOCTOR: No. No! No, I will not!
(Against his will, the Doctor turns left, stepping high, as the Great One laughs.)
DOCTOR: No! No, I will not! No!
(The Doctor has turned right round and back to where he started.)
GREAT ONE [OC]: Is that fear I can feel in your mind? You are not accustomed to feeling frightened, are you, Doctor? You are very wise to be afraid of me. Go now. You must hurry back and fetch the crystal. I must have it, don't you understand? I must have it! I must! I must! I must! Go now. Go! Go! Go now!
(The Doctor runs back down the rock passage.)
K'ANPO: The moment I have been waiting for. The moment of truth, if you'll forgive a cliche. The moment of truth for us both. You know what you have to do?
DOCTOR: No, I...
K'ANPO: I think you do, my son. What is it you most fear?
GREAT ONE [OC]: Is that fear I can feel in your mind? You are not accustomed to feeling frightened, are you, Doctor? You are very wise to be afraid of me.
DOCTOR: Is there no other way?
(The Doctor steps out of the Tardis and falls onto the floor. The Brigadier puts a cushion under his head.)
SARAH: Oh! Oh, Doctor, why did you have to go back?
DOCTOR: I had to face my fear, Sarah. I had to face my fear. That was more important than just going on living.
SARAH: Please, don't die.
And then (40-year old spoiler warning), the Third Doctor then regenerates into the Fourth.
There are plenty of things in this story that a normal person might be afraid of, such as giant talking spiders that shoot force lightning, that crystal cave with what I assume is some kind of lethal radiation, and the even more supergiant talking spider with telekinesis strong enough to control the Doctor's body against his will.
But none of these things are particularly unusual for the Doctor, and the Doctor shows no particular indication of fear (that I can recall) prior to the first quote above. I'm pretty sure he's been controlled by superpowered aliens before (though right now I can't recall a specific story where that happened), and if he had bogstandard aracnophobia it ought to have come up sooner than that.
My best guess is that he's simply afraid of death, since this is arguably the first time he's ever "died" (the first regeneration was due to old age, and the second was done by the Time Lords; neither was caused by disease or injury). But since he knows he'll just regenerate anyway I'm not really satisfied with that.
Have I missed any indication of what he was or wasn't afraid of in this story?