Units of measurement varied considerably from place to place, even on a regional level. Certain ones were more widespread because they were related to practical aspects of the daily life. For example, the acre was defined as the amount of land that could be ploughed in a day with a yoke pulled by an oxen, so it was a more commonly understood unit of measurement.
However, many were also the local units of measurement. In the Kingdom of Naples around 1130, when Roger of Hautevillle became king, there were both acres and the *moggi (*possibly called "almuds" in English), which were equal to 3364 sqm. in Naples, but 3333,33 sqm. in the counties of Teano or Capua (roughly 40 km each from Naples).

Others like palms, feet, arms were also broadly understood and accepted as the variation was not too ample and in case the need arose (i. e. buying several palms of cloth), merchants had measuring tools calibrated to the most accepted amount for the region. This was true both for length and weight. A common example I can recall are the differences for measuring the weight of precious metals in the royal Carolingian mints, where a pound of silver was to be equal to 407,4 g to mint 200 silver coins, but grain and flour were counted in "tomolii" in Italy and each city might had had its own specific weight value.

Especially for the Dark Ages we can assume that most measurements were made using either local customs or through less "orthodox" methods - like walking up and down a patch of land and making a rough estimate which I'm sure happened at some point, to the extent of seeing it done by the elderly farmers in rural parts of Italy.

There were surviving units of measurement from the Roman times that were still employed but I cannot give you an exact explanation over where and how they were preserved and subsequently used.

What we now call the "imperial" system was pretty widespread in Central/Northern Europe during the early medieval period, and thereafter. Yards, feet, inches, furlongs, miles, and so on. There were also a number of now-forgotten administrative units which were common throughout this region, in this period, the most important of which was the "hide". This was used in England, Scandinavia (especially Denmark) and much of Germany.

How do we know this? Put briefly, fields. There was a large sub-discipline of history in the early 20th and late 19th centuries that went and measured the different kinds of fields which were used for agriculture in Germany, England, Scandinavia. They found that certain patterns in the way that agricultural land was divided were constant in, for want of a better descriptor, "Germanic" regions. Whilst they took this, at the time, to imply that this region was inhabited by a single, homogeneous ethnicity of "Germanic" peoples, it's been more recently concluded to be more likely that they just shared a system of measurements, i.e. the measurements given above. We know that the conversion rates of inches-feet-furlong-mile remained pretty much constant, because legal texts from the early twelfth century in England give the exact same conversions that we use today.