Comparison-based data structure for finding items
Here's a proof that it's impossible. Suppose you could build such a data structure. Build it. Then choose $n/\log n$ items at random from the list, add $\epsilon$ to each of them, where $\epsilon$ is smaller than the difference between any two items on the list, and perform the queries to check whether any of the resulting items is in the list. You've performed $O(n)$ queries so far. 

I would like to claim that the comparisons you have done are sufficient to tell whether an item $a$ on the original list is smaller than or larger than any new item $b$. Suppose you couldn't tell. Then, because this is a comparison-based model, you wouldn't know whether $a$ was equal to $b$ or not, a contradiction of the assumption that your data structure works.

Now, since the $n/\log n$ items you chose were random, your comparisons have with high probability given enough information to divide the original list into $n/\log n$ lists each of size $O(\log n)$. By sorting each of these lists, you get a randomized $O(n \log \log n)$-time sorting algorithm based solely on comparisons, a contradiction.