Is greedy algorithm the best algorithm for set cover problem?
The result you quote, due to Feige, has actually recently been improved by Dinur and Steurer (based on earlier work of Moshkovitz), who showed that unless $P = NP$, there is no polynomial time $(1 - o(1))\ln n$-approximation algorithm for set cover.

This result states that if all you care about is the worst case approximation ratio, then you cannot substantially improve on the trivial greedy algorithm. (There are also other algorithms achieving the same $\ln n$ bound.) In particular, no polynomial time algorithm gives a constant factor approximation for set cover on all instances.