You can get an initial benchmark for the size of the ship from the largest cruise ship on Earth today:
Royal Caribbean's newest ship, Harmony of the Seas, debuts Friday on its pre-inaugural sailing out of Southampton, England. Weighing 226,963 gross registered tons with a passenger capacity of 5,479 guests at double occupancy (it fits 6,780 guests total) and 2,100 crew members, Harmony is now the world's largest cruise ship.
Thus, a cruise ship has about 26 tons displacement per person on board with quite a bit of crowding and only a week or so's supply of food, but also with fuel and engine space in addition to areas intended to be occupied (we will need to account for fuel and engine space separately in this case because the fuel and engine requirements of an interstellar ship are very different from those of a cruise ship).
By comparison, the Nimitz Class aircraft carriers are the largest warships ever built at 102,000 tons with over 6,000 personnel, which is about 17 tons per person for tours that are typically six months at a time, with less fuel (since it is nuclear powered - an aircraft carrier can run itself for about 30 years before refueling, but not no fuel because it has fuel for aircraft), but with space devoted to aircraft that would not be as necessary on an interstellar craft (although some landing craft would surely be needed).
But, you wouldn't go wrong in estimating that for a many year long trip as opposed to a weak long holiday outing or a six month tour for someone use to the hardship of a soldier, that you'd need at least 45 tons per person of living space per person.
It takes roughly 10,600 square miles of arable land with crops growing on them to feed the population of Seattle (with a population of about 652,000), while the city itself has less than 84 square miles of land area and not all of that is arable land (i.e. land that it is possible to grow crops upon). This is about 10 acres per person. And, any interstellar trip is going to need to grow most of its own food (with artificial light because starlight is too dim). Even if you could be significantly more efficient than terrestrial farming on Earth which isn't optimized for land being extremely scarce, by an order of magnitude, you'd probably need at least an acre per person of space for food production.
State of the art terrestrial farming techniques and a vegan diet leave you at about 2 acres per person. The most optimistic estimates I've seen are as little as 1/4 to 1/8th acres per person, but some of the assumptions that go into that aren't well proven or demonstrated in practice. So, an estimate of 1 acre per person is fairly reasonable middle ground.
Still, if you were really going to build a ship like this, you'd want to invest a lot in improving agricultural productivity per square meter because each percent of reduction in this reduces you ship size and cost by 1%. 947
If you can produce enough food per person with 1/2 an acre instead of 1 acre, you can cut the size of the ship in half.
A 102,000 ton aircraft carrier has about 4.5 acres of deck space. So, you'd need about 22,700 tons of food production area per person in addition to the 45 tons of living space.
So, in round numbers you'd be talking 23,000 tons per head of living space and food production space and nuclear power production for ship life support operations from which to feed and house them assuming an order of magnitude improvement in food production per square foot relative to Earth would be possible (e.g. by reducing less efficient animal food relative to more efficient plant food proportionately).
I'll assume that "millions of people" means 2 million, for sake of having a number to work with here, so you'd need at least 46 trillion tons of living and food production space (which would still be quite cramped) before considering fuel and engines.
You can make your ship as big as you desire, because for very large interstellar space ships, the amount of fuel and engine required per ton of living space is going to be nearly constant.
There is another moving part here. More fuel can allow you to travel faster, less fuel limits your speed (engine size is pretty indifferent as the key question is how long you run them at an acceleration of 1 G, not the amount of peak acceleration you can generate). For example, if you decided that 95% of the ship would be devoted to fuel and engines for a near maximal peak speed given the available technology (whatever that ends up being), then you have a ship carrying 2 million people that is at launch 1 quadrillion (i.e. 10^18 kg) of displacement. An aircraft carrier by comparison is 10^8 kg of displacement. So, this ship would need to be on the same size as about 10 billion aircraft carriers. 947