A quick search at the Navy website indicates that there are no battleships currently in service with the Navy. I told my son that as I understood it, the Navy had concluded that aircraft carriers are better able to perform the tasks the Navy wants to accomplish than are battleships. He wasn't happy with the answer, but I left it at that instead of trying to convince him with lots of statistics. I think that I can say that his reasons for preferring battleships are sentimental rather than utilitarian. Fair enough, he's eleven. And it seemed enough of an event to hang a post around.
For some context while avoiding all the maths, in Book 2, the biggest ship size possible was 5,000 displacement tons. I personally never used a ship that size. This makes the largest ships about half the size of the RMS Titanic, and a third the size of a modern aircraft carrier. Big, but certainly not the mammoth sizes that Star Wars impressed upon everyone's imagination. Under High Guard rules, it was possible to design ships 100 times the size of the biggest "Book 2 rules" ship. Now that's a big ship! The problem with this is that now the PCs are very small fish in the big pond that is outer space. See this photo mashup from Jeff's Gameblog for some sense of scale:
The PC's will be flying ships 1-3, maybe. Ship 7 (in white) is a High Guard designed ship. See the difference? With High Guard ships, you can fight gigantic fleet actions without having to have thousands of ships involved. In the big setting of the Third Imperium, having these big ships makes sense, as the Imperium can certainly afford them. But it leaves the PCs very much on the margins. The High Guard rules make it virtually certain that in a big vs small encounter, the small will get smashed, and the big will be unscathed.
In My Traveller Universe, I've tried to chart a third course. I use both Book 2 and Book 5, but I keep the two kinds of ships separate. The PC's ships are defined by Book 2, and I only use Book 5 for strategic fleet actions between the navies of rival star nations. If for some reason a Navy ship gets into a fight with a Book 2 ship, I convert the Navy ship to Book 2 stats and run with it. Adventures can happen in big ships, the canonical example for Traveller is Adventure 4, Leviathan, where the PCs are crew/officers aboard a large merchant explorer vessel. But most of the time, PCs are going to be in small ships that only need a few people to operate.
Starship combat is a fun game in its own right - Book 2 rules are very modular, it takes just a few minutes to work up a ship. I've 'house-ruled' the system to include elements from Book 5 like armor and the advanced weapons, and I'm looking to incorporate the variant damage rules from Space Gamer #40 (Traveller: New Space Combat System, by Steve Winter).
The vector movement system employed in Book 2 is realistic (for a 2D model of space) and requires forethought and planning, as there is no 'turn-on-a-dime' available. The selection of weapon load is often critical, and the computer rules require careful allocation of resources. Computers have limited processing power, and while you may have lots of storage space to hold programs, you have to determine turn by turn which ones are most important to have running. I know much ink and many pixels have been spent on pointing out the flaws in Traveller's computer architecture, but the limitation of only being able to use so many programs at one time adds to the challenge.
Whether large or small, starships play a big role in Traveller - they are what allow the PCs to travel.