[a humble lich]

I'd say it varies by quite a bit. When you say the middle ages, you are talking about a continent sized area over a time frame of a thousand years. There were regions with a strong feudal structure like France and England. There were republics and free cities like Venice, Hamburg, and Switzerland. There were border regions like Prussia, Ukraine, or Spain. There were remnants of old Roman law. And there was also variation over with time. Acquiring land in 1500 was very different from acquiring land in 600 (when to become landed you needed to get some troops, take some land, and fight anyone who wanted to take it from you).

I would imagine at some times and places to acquire land one would need for a noble to give it to you in exchange for feudal oaths. In more republican areas one might be able to just buy it. In some lightly populated areas one might be able to just go out and find unused land. Another common method is being useful in a war and being given a title for newly conquered land.

It also depends on what you mean by owning land. In a strongly feudal area, the only one who owns land is God. God then grants the king rights to administer the land, who then gives rights to dukes, who then give rights to lesser nobles, who then let commoners use the land. In theory, at each level there are taxes, fees, and rents which must be paid as well as privileges, responsibilities, and duties.