Yes it is possible.
For the Vice President there is a specfic example of Andrew Johnson, a pro-union Democrat, who formed a "National Union" ticket with Republican Abraham Lincoln.
There is no special requirement that the Secretary of State is from the same party, although there is an expectation that Secretaries of State will promote the policies of the administration, and so most Presidents will choose someone who they can work with politically. Some Secretaries of State have been seen as moderates with links to both parties (Colin Powell for example has been touted as a Democrat VP nominee, before siding with the Republicans in the mid 90s), however I can't find an example of a President hiring a political opponent to this role. (Another answer mentions Democratic President Grover Cleveland's Secretary of State, Republican Walter Gresham, which was a good find, however by the time of his appointment Gresham was actively supporting Democrats, and had campaigned for Cleveland in the Election. So he was hardly a political opponent of Cleveland.)
And below the Secretary of State, it is possible for a President to appoint people without a strong political position, or to maintain their predecessor's appointment for the sake of continuity. An example would be Robert Gates, who served as Defense Secretary in both the Bush and Obama administration.