Clarification on NP-hardness and hardness of approximation results for set cover?
When you have a result like "It is NP-hard to approximate X within ..." or say "X is NP-complete", it means that there is an infinite family of instances of X that are hard. This is what worst case intractability means.

In your example, the hardness of approximation result holds for the set cover problem, meaning that there is an infinite family of instances of set cover that are hard to approximate.

At this point, you might put further constraints on a problem in the form of "suppose every set satisfies this property" or whatever, but this is then a different problem, i.e., you are narrowing down on the instances that you care about. If you have a statement that says "set cover with your favorite property is NP-hard to approximate", it means that there is (still!) an infinite family of instances of this problem that remain hard.