This was the spirit of Churchill's nominations as Prime Minister of the United Kingdom during WWII in order to unite the country.
The two biggest wars since the founding of the United States (US) were the Civil War and World War II (the Revolutionary War was obviously before the founding). World War II occurred during a period when the Democrats were incredibly dominant. They didn't need more Republican support. And of course World War II was not as significant to the US as to Europe or China. Other than Pearl Harbor, the US fought World War II outside the US.
It's not entirely wrong to regard the Civil War as being between the anti-slavery Republicans and the pro-slavery Democrats. Even if Abraham Lincoln had offered, it's unlikely that there were enough Democrats interested in being in the war time cabinet. And if there were, without the Southern states, Republicans dominated Congress and could have blocked such nominations. In 1864, they did create a National Union ticket of Republican Lincoln and Democratic vice presidential candidate Andrew Johnson. Edwin Stanton was a Democrat but switched to being a Republican to become Secretary of War in 1862.
Outside a war, an even division of Democrats and Republicans would be unlikely. Recently there have been some efforts to pick one or two cabinet secretaries from the other party. For example, Barack Obama retained Bush's Defense Secretary, Robert Gates, and almost hired Judd Gregg as Secretary of Commerce. Robert McDonald of Veteran Affairs and Ray LaHood were also Republicans.
You can see a full list of cross-party appointments on Wikipedia. Most of the appointments in that list were to lower level positions. There was never an evenly split, unity cabinet. The closest in approach was the Lincoln/Johnson National Union ticket of 1864, but that didn't carry over to the administration.