I see it as being quite common. Primarily my own career so far has been split between three large multi national companies that all had graduate programs. There are also always entry level roles that will get you into a company and once you can show that you can perform and hit your goals, there will nearly always be room for you to grow within the company.
There are many possibilities for a career with a bachelors degree! For government careers, you may want to consider an internship. It would be very helpful to know what area of government you are interested in?