I don't recall if any story has ever "scared the socks off me", but a few short stories I've read over the years have had significant emotional effects on me.
I've realized that I have a mild phobia to rogue bodyparts and stories that contain them. Both Whitehead's The Passing of a God and A Planet Named Shayol (nominally SF but with a firmly horrific element) made me feel distinctly queasy both during and a little after my reading of them. Likewise, stories that contain killer midgets/fetuses (quite a niche genre I'd assume) give me the willies: Henry S Whitehead's Cassius and Jean Ray's My Cousin Passeroux being notable examples, the latter story being one of the creepiest horror stories I've ever read.
Several pieces have a depressing effect on me. Most of M John Harrison's stuff, and quite a bit of Joel Lane's as well. Ligotti's stories tend to linger with me for a while after reading, making me see the world in a slightly more bleaker tone. Robert Aickman's stuff makes me feel similar, not depressed so much as thrown out of kilter. It's not a pleasant feeling, but then again it isn't unpleasant either.
I'm not his biggest fan but Ramsey Campbell's excellent short story, Mackintosh Willy, has stayed with me ever since I read it, particularly one scene involving a radio and the static-filled voice that comes through it. Both Laird Barron's and TED Klein's superb novellas The Procession of the Black Sloth and Black Man With a Horn perfectly ramp up the dread and weirdness its protagonists suffer to a fever pitch.
I didn't used to think it possible, but a few pieces I've read recently have actually made me jump, in the pleasant way a good slasher flick can do. Jean Ray's two masterpieces, The Shadowy Street and The Mainz Psalter have scenes that made me feel besieged and on edge, a combination of alternating scenes of calm strangeness and brutal sudden violence. Edogawa Rampo's The Human Chair has, despite its horribly disappointing ending, one of the best, simplest and most horrible ideas in horror fiction, and it's executed wonderfully. Try reading it whilst relaxing in one of those big sofas they have in coffee shops.
The Willows and The Wendigo by Algernon Blackwood, The House on the Borderland by William Hope Hodgson, Lovecraft's At The Mountains of Madness all inspire that wonderful awe of being exposed to a vast and utterly alien other world. In its own way, this is horrifying.