Is it true the the serfs were not able to own the land they farmed?

Serfs did not own land. They lived and worked on manors owned by members of the nobility.

2nd answer: That is only partially accurate. Serfs did not own land in the modern sense. But many did have rights to a certain amount of farmland in the field of the manor they resided on. They owed the lord some combination of rents, fees, and labor for this land, but by tradition and practice a serf could not be deprived of the land he held from the lord. Serfs were not slaves. They could not be bought or sold, or forced to leave their holdings. They were not fully free either, as they did owe an amount of labor to their lord, as well as rents and various payments in kind.

The lord of the manor did not really "own" the land of the manor either. The lord might hold this land from a greater lord, and in exchange owe military service or money in exchange. Much like the serf, the lord of the manor could not be arbitrarily deprived of his holding, but he had duties and responsibilities regarding it.

So for any given piece of land, there were several people who had both some claim and also responsibility relating to that land. There was very little if any "fee simple" property, to use the modern legal term, in the middle ages, meaning property that was clearly owned by one person without outside duties or encumbrances.