You need Hull Sizes as techs. Research the Hull Size tech Large Building for Battleships

It is an Orange Technology in TA - you should be able to go to the Research Screen and follow the tree there and it will give a description as to what it will let you do.

If you have this Technology and still cannot build Battleships, then there is something else wrong.

Full games are completely different. In the campaigns you have a specific goal and nothing else matters, it is a mistake to try and build an empire if playing a campaign.

The full game is about an empire. You have to balance research/economy/military and you have to exploit diplomacy/trade/and tactics. There are resources to harvest, anamolies to explore, borders to maintain, people to appease, the list goes on and on. Personally, the full game is where the excitement is, but the campaigns are neat to see how the story develops. If you are interested in the story of GalCiv2, then the campaigns are a nice way to immerse yourself in it. If the story is all you care about in regards to the campaigns, then play them on Cakewalk level just to see the story develop.

Note: A frequent problem with new players when playing the campaigns is that they try to play them as they would a regular full game. This leads to frustration. Read carefully what the objective is for every campaign, and concentrate on nothing but reaching those objectives.

Enjoy!