For me, rich is less of a dollar amount than it is the capacity to think of money in a different way. Let me explain.
Currently, I am a student. I have massive debt, I buy only the things I absolutely need, and I am forced to make monetary decisions based on available resources.
"Rich" to me will mean no debt (excepting a mortgage, for awhile), no reason to worry about finances to purchase the things I want, and the capacity to travel when it suits me. A nice retirement and investment portfolio is also necessary, but you don't have to be "rich" to accumulate good retirement savings or investments.
I made a list once of all the things I ever wanted. At the time, it totaled to something like $78,000 but it excluded a house or a car or regular annual expenses (health insurance, car insurance, etc.). It did include, however, all of the tangible things I ever wanted.
In summary, being rich is a state of mind when money is no longer an object or a number, but a background component of one's life. In my future profession, I have little doubt that I will obtain all of the "things" I want, but I am more looking forward to changing the way I view money entirely.