Hash table

Is it ever possible to have a Hash table that can do get,put,remove @ O(1) for Strings?

calculating the hash for the String alone is O(n), is it not? [that's if you want to produce a good hash that won't collide all over the place]

and a hash set could be represented using an infinitely long bit string, where the ith bit is on if and only if the ith string is in the hash set.