With modern technology, can we build a ship like those in the movie 2012?

There is nothing that appears impossible.
For being as big as it is, the ships has incredible bouyancy, judging from the art in the first picture. A tanker that size would sit much lower.

It might be that the interior bulkheads are few, or that it has a lot of volume without heavy machinery. It looks to be conventionally powered. But there are big problems with the concept, if you are attempting to preserve other forms of life than human, and the apparent size is about that of the Eiffel Tower in length, say 1200 feet. Can you provide enough structural integrity for such a size? Yes. The Oasis of the Seas is 225k GT, and it is over 1000 feet long and 200 feet wide. Max capacity is about 7000 people. The scale is hard to grasp until the shots showing people against the hull; it still is hard to estimate, but it is a bit larger than the largest cruise ships, say the Knock Nevis (1500 feet) size times two high.

To give you an idea of scale, here is the Batillus tanker, and it is about the same size to my eye:

The largest floating object is the Prelude (Prelude FLNG - Wikipedia), sitting off the West Australian coast. It has a displacement of 600,000 tons, or is five times the displacement of a US aircraft Supercarrier. This would seem to be around a similar displacement of the arks in the 2012 movie.

The largest vessel the world has ever seen

Worlds largest floating LNG facility under tow from Korea to WA

Building a cruise ship with that kind of size would put it around being able to accommodate 22,000 people given current cruise ship density. Cruise ships aren’t overtly tight, with large accommodation areas and many entertainment and other areas, but would given an idea of birthing for the size of the ship.

Most cruise ships a limited in size by the ports they can dock at and the movement of people on and off. Also most cruise building yards are designed around ships being a certain size. Obviously with an ark type ship, this would be less of an issue, and a larger size could be built. Certainly we can build ships of this size.

Making it able to withstand a wave that would splash over it would be significant, but do able. The bow reminds me of some of the off shore wave piercing designs, which have started to appear on some military auxiliary vessels and tankers.

So yes, possible. Certainly a unique design, but plausible with current technology.

I would argue the ships are the most believable part of the movie