Hi Heesob, "ме е страх" means that you are afraid, but in the sentence above is used also "не" infront of "ме е страх".Because of that "не ме е страх" = I am not afraid. More for the personal pronouns you can find here: http://www.slav.uni-sofia.bg/grammar/morf16.html

That's a very interesting question! Yes, that's the most natural way of saying "I'm afraid", but you should rather think of it as an exception, not as a rule - there are very few nouns which can be used that way (Actually, I can recall only "страх", "срам" and "гнус"). It seems to me, it's related to the verbs, these words are usually associated with (i.e. "обзема ме","обвхаща ме"), but that's just a wild guess, I don't know for sure!

В съчетание със съществително и местоименията ме, те, го или наречие и местоименията ми, ти, му означава състояние, чувство или усещане според значението на съществителното или наречието.