What makes Jim’s heart sink “into his boots”?

That would be the island............

Perhaps it was this—perhaps it was the look of the island, with its grey, melancholy woods, and wild stone spires, and the surf that we could both see and hear foaming and thundering on the steep beach—at least, although the sun shone bright and hot, and the shore birds were fishing and crying all around us, and you would have thought anyone would have been glad to get to land after being so long at sea, my heart sank, as the saying is, into my boots; and from the first look onward, I hated the very thought of Treasure Island.