A rational actor is one who is able to decide what he thinks about an option, and who makes decisions that are transitive (ie. consistent). In that sense, rationality is still the basis of economics, but this is not what a normal person understands as rationality.

Instead, a more narrow view (partly due to difficulties of dealing with uncertainty, partly due to the context of preferences in a dynamic setting) required adding other behavioral theories which are now rightly considered mainstream in economics as well. All these non-rational actors are still rational in the general sense, in that they make choices based on preferences at some point in time.

There is, by the way, no other science dealing with human behavior that does not assume rationality. In business, this is called "bounded rationality" after Simon, but nowadays is basically and excuse to use rational actors even though we know "they ain't". All sociological actors are rational as well, even though you will rarely find a sociologist write this as a statement. Still, choices are made as responses to the environment of social forces.

Quite generally, it would be impossible to theorize about human behavior if it were irrational. That, in the general sense, would mean that it is not consistent and not predictable and thereby inherently not accessible to science.

What you econ Professor told you is basically correct, but the correct formulation of this you will probably see the first time when you take a course on decision theory or graduate microeconomics.