Well, I've worked at the federal level for about eight years so I don't think I'm an expert, but I'd have to say that it depends on the department for which you work. Working for the U.S. Fish and Wildlife Service will be different than working for NASA. One is populated mostly by biologists, and the other mostly by engineers. However, I will say that the federal government like many other publicly-funded entities is not as welcoming to very ambitious or enterprising individuals. This may eliminate the "E" types, as well as other types to expect things to happen quickly. I worked for a scientific organization and was surprised at the sometimes complete lack of logic when designing projects, etc. If you are intolerant of inefficiency and downright absurdity as many TJs would be (myself included) then navigating the politics and poor organizational management that you sometimes find in government could make for a very toxic situation. Sometimes efforts seem to be a bit of a farce or "boondoggle". They are not affected directly by market forces, so bad management is often times accepted or even the norm.
Having said that, there are many government organizations that do great work. Again, it really comes down to the department, agency, and even office in which one works. Also, many introverted types may find government to be a safe harbor and even thrive.
President ENFJ. (If it were a for-profit corporation I would go with ENTJ, but running a government fairly requires attention to the way people feel and ability to merge disparate perspectives for compromise.)
As for his cabinet, a bunch of INTs with a few INFs thrown in. They can help with the more technical problems and advise the president on what to do, as long as they stay out of the public eye.
Vice President ENFP or ESFP, some kind of sunny face to help keep the public happy and advocate for the government's new ideas, and make the people feel like they have someone in office that they can really relate to.
If you could be anything you want, I bet you'd be disappointed--am I right?