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If I'm a peasant during the dark ages, what surface (in meters (<- I am in advance on my time)) I need to farm to get enough cereal (for bread and brew) ?
how many cereals (in kg (<- again, I'm in advance on my time) this surface will deliver per year ?
In Carolingian times, the yield of grain on average soil was 2:1. For each seed planted, you harvested two. Starting with the eleventh century, an upward trend brought agricultural productivity to an average of 4:1. This meant 8-12 bushels (200-300 kg) of grain per acre. Let's just conclude that in Carolingian times, in Western Europe, an acre of land gave 4-6 bushels (100-150 kg) of grain. Since it isn't clear whether the absolute yield increased or merely the seed-to-seed ratio. It might as well have been that the average Carolingian plot gave 8-12 bushels per acre, and half of it was kept as seed. Let's go with the second hypothesis, since an absolute yield of 4-6 bushels would have meant 2-3 used for consumption. Let's say an acre gave 8-12, and 4-6 were replanted the following year, whilst after the agricultural renaissance of the eleventh century, only 2-3 were replanted, leaving 6-9 for consumption.
An average human being required back then 14 to 24 bushels of grain per year to survive. Let's say children were situated at the lower margin, whilst full-fledged, hard-working adults at the upper one. Therefore, 4-6 acres were necessary to sustain an individual. Your household comprised of 2 adults and 2-3 children. You would have needed then 72-86 bushels per household - 12-22 acres. But then, again, I suppose that as absolute yield 8-12 is pretty high for the Dark Ages. I'd go with 6-8 bushels per acre gross yield, 3-4 net. This leaves a necessary of 6-8 acres per individual, circa 30 for a household. In the High Middle Ages, it is well known that 10-15 acres of land were necessary in order to sustain an average household.
Later edit: Oops! Sorry, I forgot to take into account crop rotation. Around half of the land was left fallow. The consequence of this is that even if in absolute terms 6-8 acres produced enough grain in order to feed an individual, his long-term needs of land were doubled - 12-16 acres (~5-6 hectares). Thus, an average household could have required even 50 or 60 acres (~20-25 hectares) of cropland.
References: "Life in a Medieval Village" by F. & G. Gies - a very accessible book, price- and content- wise, yet not lacking depth.
"Rural Economy and Country Life in the Medieval West" by G. Duby is a very thorough, sometimes husky piece of academic toil. (Pretty much all of the Annales School is obsessed with demographic and ecological macro-historical trends.)
"Early Medieval Settlements: The Archaeology of Rural Communities in North-West Europe, 400-900" by Helena Hamerow
"Peasants in the Middle Ages" by Werner Rosener
"The Agrarian History of Western Europe" by Slicher van Bath
"Peasant Life in the Medieval West" by Robert Fossier
"Medieval Farming and Technology: The Impact of Agricultural Change in Northwest Europe" by Grenville Astill and John Langdon
About 0.25 wallach was enough to support a family (so at least two adults and all they children) on a good land in Lithuanian territory at about 1600. The size of wallach was about 21.3 ha so 21 3000 square meters or 52.6 acres. Wallach itself was a norm for a relatively easy, decent life. Family was a team, and all had they specialized work roles, including older children. Probably none could be as efficient with the proportional part of the land alone.
Just enough area to grow crops is not enough because crop rotation is required when using medieval technologies only.
It highly depends on what you grew on it, where, and how many tools you had at your disposal.
Generally vegetables had higher yields, but were harder to harvest. So the reason why grain became a staple food is that there was just enough place for it to be grown. Legumes were also a staple food; they provided more calories per hectare besides bearing much needed proteins. Crop rotation required that one in three fields lay fallow, but farm animals could pasture on that providing alimentary diversity.
Crop yield was much better in southern Europe than in Northern Europe, especially in fertile river valleys. You could grow more wheat, cut grass more often (more hay), harvest crops more often and grow better/less cold resistant variations of staple crops. In the north you had to rely on barley and rye instead of wheat, later on potatoes, rice and maize improved the calories per hectare ratio both in the south and in the north. In the south they had olive oil too, which was a valuable resource, because it had high calory yields and relatively low soil requirements.
Equipment, training, access to technology and further food sources and tax policy of your local lord also play a big role. Of course you had to buy and sell goods to pay your taxes and to buy new tools. If for various reasons, you were not a serf, tax might have played a smaller role. For example a yeoman in England or a "Freibauer" in Tirol were exempt from feudal taxes. Living near monasteries often provided an extra help regarding agricultural technologies, as monasteries also spread that. If you had farm animals you had to do much less work yourself, saving a lot of calories yourself too.
Finally, where i come from (the Alps) in the 19th century 3 hectares were enough for subsistence and a small extra for a family (mother, father and somewhere between 5-20 children usually employed on the farm) if you had a good farm on the valley floor. High pastures and the like usually required several dozen of land. As much of the harvesting technology on high pastures and their living conditions now are pretty much the same as in the middle ages, except for better cow breeds, they are a a good indicator for what a family might have needed. Taking into account that animal husbandry requires a lot of space, people were smaller and living standards lower, cows gave less milk and that you had secondary sources (fishing in lakes, hunting in forests during winter) - i guess a family of yeomen not living in high altitudes with good soil and a good location, probably lived off less than 5 hectares.
EDIT: though as a rule of thumb crop yields are better in southern europe's climate, one has to take precipitation into account, i didn't. Also some of the north eastern parts of europe are rich in so called black earth, which is like a gold mine for farmers.
If it's said that yield of an acre are 8 bushels - your acre will produce 8 bushels working in three field system. So they will need 70 bushels for food (5 member) and 18 for seed = 88 bushels. One acre 8 bushels = 11 acres. (in eleventh century +)
