Purpose:
To determine the ideal farm size to maximize yields with the least amount of labor input by altering both the farm size and worker number (including using more or less workers than the in-game default).
Methodology:
All Farmers were educated with steel tools, with full health, and maximum happiness. I paused the game and inspected each worker to verify this each year during the early-spring.
All of the homes for the farmers were directly touching the road next to their farm. I used the pathing tool to verify that the farmers were using the proper home closest to their farm each year.
Each farm had its own storage barn directly touching the road next to the farm. I followed a random sample of workers each year to verify that they were using the closest barn and not wasting time running around the map. I didnt reassign any jobs or build anything during the tests to ensure that homes werent being swapped around or workers shifted around based on proximity to the new work/buildings.
All crops must be completely gathered by the middle of autumn otherwise I consider the farm to be underworked, which would increase the risk of losing crops due to an early frost.
All storage barns are inspected before each harvest to verify that they weren't full.
I had one-third of my total population (~125 people) set as laborers year-round to minimize the chances of my farmers walking across the map to do laborer jobs during the off-season (i.e., winter) and then end up getting a late start in early spring planting the seeds.
I built three of each tested farm size and ran the tests for 5 years and averaged the output. More thorough testing than this is mostly unnecessary in this game because theres no variance in the year-to-year yield aside from the variance caused by weather and villager pathing, both of which i attempted to control for.
All tests were done using corn, which is the second fastest growing crop after beans.
Tests:
4x14 (56sq) w/ 1 worker: Done by the beginning of late summer; 392 average yield; Conclusion: Heavily overworked, farm size too small. This is the farm size I've seen recommended often here on Reddit (any farm size that's a multiple of 56 squares using the default number of workers), but as you'll see below, it has an output 2-3 times worse than other farm sizes where you manually select fewer workers than the default.
8x14 (112sq) w/ 1 worker: Done by the end of early autumn and the beginning of autumn on average; 784 average yield; Conclusion: Farm size is close to ideal, if space isnt a concern farm size could probably be increased slightly which would increase yields during good weather without significantly reducing the yield during bad weather.
11x11 (121sq) w/ 1 worker: Done by the middle of autumn, usually right before the first frost of the season. Lost ~20% of two farms due to an early frost one of the years; 868 maximum yield, 845 average yield. Conclusion: Closer to ideal than 8x14. Farm size could be increased slightly to increase yields during perfect weather.
13x10 (130sq) w/ 1 worker: Done by the middle of autumn to the beginning of late autumn, lost ~20% of the total yield one year to a mid-autumn frost; 924 maximum yield, ~740 yield during the frost year, average yield of 887. Conclusion: Higher maximum and average yield than the 121sq farm but the risk of losing crops to early frost may outweigh the potential benefits. With a less than perfect setup (e.g., with long walking distance between homes and work) the risk of loss to early frost increases.
15x15 (225sq) w/ 2 workers: Done by the beginning of autumn; Average yield of 798 per worker; Conclusion: Close to ideal but still less output than the 1-man farms. See overall conclusion below.
Overall Conclusion:
The ideal farm size for corn production is somewhere around 121 squares per worker. The exact size doesnt matter since farm output is always ~7 food per square, so choosing a farm size to fit your available space while still being within ~10 squares of 121 would be ideal (so between 110-130sq per worker) since it would allow you to maximize your usable land while not compromising total output by too much. The distance the worker has to walk between their home, their work, and the closest storage barn should also be considered. The less optimal their walking path is the smaller the farm should be. As a very rough rule, if the house and barn are touching the farm, size the farm to ~121sq and for every five squares the house and barn are away from the farm reduce the size of the farm by 1sq. So, as an example, if the house is 10sq away from the farm and the barn is 10sq away (so a total of 20 additional sq), reduce the farm size by ~4sq to 117sq.
A 15x15 farm (225sq) being worked by two workers would have an output very similar to a 112sq farm being worked by one worker, but the total potential output would still be less than two 1-man farms of 113sq-130sq. A 15x15 farm is also very difficult to place compared to a farm with ~121 squares since you can choose from multiple different orientation (i.e., 8x14, 9x13, 8x15, 10x12, 11x11, 9x14, and 10x13) to fit your useable space. The only advantage a 2-man, 15x15 farm has over a smaller 1-man farm is it takes less time to change the crops and disable work if needed, which would make reassigning your farmers during the winter easier.
TL;DR:
To maximize the output of your corn farm, choose a size between 110-130 squares (i.e., 8x14, 9x13, 8x15, 10x12, 11x11, 9x14, and 10x13) and use only one worker. 121sq seems to be the 'safest' size with virtually no risk of losing crops to frost while still nearly maximizing output. Only use 130 if you're in late-game with a perfect setup (i.e., houses and barns touching farms) and have a lot of food security and can risk an early frost. 110-115 is ideal for early-game (to remove risk of frost loss) and should also be used if your farmers have a sub-optimal walking distance to their farm.
Addendum: Something to note that i forgot to mention: the best way to maximize crop yield with the least amount of total labor time is to heavily micromanage each individual field based on the weather. I haven't thoroughly tested this so i can't really say for certain what the best approach is, but it seems like the ideal strategy is to use only one worker until late-summer/early-autumn on a 15x15 corn/bean field to plant the seeds and then swap it to two workers in late-summer/early-autumn to harvest everything before the frost hits. Once frost hits you take all of the farmers off the field by clicking the stop work button for the entire winter and assign them to do other stuff during that time (like mining or gathering). You could even swap to three workers in autumn if the temperature is below ~40 degrees and you don't think your two workers will finish harvesting before the frost. That way, you basically only have 1 worker on the field for spring and summer and 2-3 in autumn and none in winter. This might be one of the best approaches for early-game to maximize total resource production with the fewest number of workers but it's definitely too micro-intensive late-game to be worth the hassle as far as i'm concerned.