Yes, women can be in places of authority in the government.
At the time of writing this article (9/10/08), Sarah Palin is the Vice Presidential nominee for the Republican Party here in the United States. She is presently the Governor of Alaska. Previous to that, she was the mayor of Wasilla, Alaska. As she moves through the ranks of power and authority in American government, more attention will be drawn to her gender, especially since she has the potential of being President should her party win and something happen to McCain while he is in office.
But, Biblically speaking, there is no problem with a woman being in a position of authority in the government or--for that matter--in business. The New Testament restricts a woman from having authority over men in the Christian church (and the home), but there is no prohibition against women being in authority anywhere else.
The Bible says that a woman is not to be an elder or pastor since both offices have the requirement of being male (1 Tim. 3:2, Titus 1:6). Also, 1 Tim. 2:11-13 says, "Let a woman quietly receive instruction with entire submissiveness. 12 But I do not allow a woman to teach or exercise authority over a man, but to remain quiet. 13 For it was Adam who was first created, and then Eve." The New Testament clearly restricts the authority/teaching position in the church to qualified men. For more information on this, please see section: Women In Ministry
Women in authority over men in the Old Testament
In the Old Testament, Miriam was a prophetess (Exodus 15:20) as was Huldah (2 Kings 22:14) and Noadiah (Neh. 6:14). Prophetesses from God spoke with authority and were ordained by God that they did so. But, most pertinent to the topic at hand is Deborah, who was a judge in Israel (Judges 4:4-5). Deborah exercised governmental power over the nation of Israel. She was most definitely in a place of authority over men in government, and there is no condemnation of her being a judge found anywhere in Scripture.
But, some might ask, if Deborah was in a place of authority in Israel, why can't a woman be in a place of authority in the church? First of all, the Bible restricts the office of elder and pastor to the male (see verses cited above). Secondly, being a judge in Israel is not the same thing as being a pastor or an elder in the church. They're different contexts.
So, Biblically there is no problem with Sarah Palin being Vice President or even the President of the United States should that ever occur.