What was the size of surface of a cereal crop needed per man per year during the Dark Ages in Western Europe?

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 +)

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.

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.