Is the exact cover problem NP-hard when there is a restriction on the size?
Yes, it is still NP-hard.  In fact, it remains hard even if you replace $\log_2 n$ with the constant 3.  This follows by reduction from 3-dimensional perfect matching.

3-dimensional perfect matching is the following problem:

Input: disjoint sets $X,Y,Z$ such that $|X|=|Y|=|Z|$; a set $T \subseteq X \times Y \times Z$

Question: does there exist a perfect matching?  In other words, does there exist $S \subseteq T$ such that $|S|=|X|$ and the union of elements of $S$ is an exact cover for $X \cup Y \cup Z$?

From this statement of the problem, it is clear that 3-dimensional perfect matching is a special case of your problem: take $U=X \cup Y \cup Z$ and $C = \{\{x,y,z\} : (x,y,z) \in T\}$.



In contrast, if the size of every set is at most 2, then it can be solved in polynomial time, using algorithms for finding a perfect matching in an undirected graph.