In the Mass Effect Series, the Quarians have a population of 17 million individuals and they get all their food from 3 massive "liveships." I'll round this down to 5 million individuals supported by a single farming ship. Assuming the ship is only dedicated to the growing of crops, how large would the ship have to be in terms of available area to sustain this population?
Any currently invented farming-techniques to save on space are acceptable.
Traditional farming requires approximately one acre per person (much like E404's calculations), so we can take that as the upper limit. Of course you can use far less efficient methods and need even more space and energy, but in this instance we are actually trying to minimize our inputs.
Various forms of more intensive farming are possible. Using multi cropping and gardening techniques you could probably cut the needed land in half. In the Second World War, urban dwellers were urged to grow "Victory Gardens" to supplement their diets and release agricultural output for the war effort, and you can grow a very successful Victory Garden in a typical suburban backyard or even a balcony.
Hydroponics and related techniques like Aquaponics are even more space efficient, with suggested yields of up to 200% more than comparable agricultural practice. Aeroponics, which involves spraying the roots with a mist of nutrient rich water, rather than simply immersing the roots in water, is sometimes advertised to be twice as efficient as Hydroponics. This means you can theoretically grow 4X as much in the same space (or alternatively need 1/4 the amount of space per person).
Even more savings can be achieved by going to algae, which is a much more efficient crop, not using energy to produce stems, cellulose or other indigestible matter. The most extreme figure is a single human can be sustained by algeaculture using only 6 litres of water per day, so a fairly small jerrycans of algae solution exposed to sunlight is all it takes.
More information and figures can be found on the ever handy Atomic Rockets website, especially here: http://www.projectrho.com/public_html/rocket/lifesupport.php#id--Closed_Ecological_Systems--Growing_Plants--Algaculture
According to this question on Gardening SE, a minimum of about 0.02ha is required for four people on mars.
That's 50 square meters per person on an enclosed farm in space.
For 5 million:
50 * 5 000 000 = 250 000 000 meters square = 250 km sqare
One assumes that any additional space in the ship, below the farming deck, would be used for water storage, crew quarters, equipment, etc.
Artificial light might also make it possible to place several layers on top of each other, dividing the area required by however many layers are used (assuming that one has some sort of powerful power source aboard the ship.
Earth's land area is about 57,500,000 square miles. About 26% of this is for pasture and 11% is for farming. I assume you want to consider the 11% only.
This 11% serves the 7,300,000,000 people of Earth. The average population of a country is very roughly 35,000,000 people.
Based on these ratios, you'd need 30,325 square miles per 35,000,000 people. Or 4,332 square miles for 5,000,000 people.
Assuming you employ the full gamut of space-saving techniques, like hydroponics and anything else, you could possibly reduce that by 80% to 866 square miles.
Space to grow isn't a big issue here if we talking about 5kk people.
Heat dissipation in space is more issue here.
It needs something around 10kW plant spectrum light 24.7.365, space 50-100 square meters, volume 100-200 cubic meters per single person. Not super duper food for someone, kinda minimal setting.
The process is not much efficient and probably needs more energy.
To dissipate 10kW heat at 300K from the hull of the ship needs roughly 20 square meters.
For 5'000'000 peoples it has to be 100kk m2 surface of the hull, where heat dissipation is happening.
If they have good energy sources, like thermonuclear reactors or such, they can cool internal volume more efficient. 600K, 2 times higher is the temperature of the shell, 4 times less radius or edge of the cube, to irradiate the same amount of heat energy. It's may be a good idea to place reactor outside of the ship, this way it can work at a higher temperature.
The ratio of internal volume to the surface can be arbitrary, by scaling one of the axis, or changing form, like pencil-cylinder.
Plants do not need gravity to grow, not sure about all of them, but those which were tested in space. (I do not know is it important for trees, fruit trees or not - maybe not or not so much)
For 17 millions of people ship size is $\small \sqrt{17/5} \approx 1.84$ times bigger, and volume will be $\small (\sqrt{17/5})^3 \approx 6.27$ time more per person.
So as you may see, the bigger ship is, the less problem is the volume or internal surface.
