Depend where you are and when you are. So lets talk about norfolk around south creake. At the time of the doomsday book (~1086) you had 5 freemen out of a working population of 37. So 5 tenant-farmers with everyone else further down the social scale. The distiction is that freemen paid rent where as villeins and below had to provide labour.

However the thing you see very quickly is the breakdown of the fudal system with pretty much all the land switching over to rent and even what villein land remained being traded and the labour effectively being treated as a form of rent.

By 1475 smallholders owned 42% of the land in the area but this actualy falls away with small holders holding just 17% in 1559. So yeah it varies a lot by time and place.

As for what a non-aristocrat could own, a freeman could own land basicaly anywhere they could find someone to sell it to them although it might come with some pretty heavy restrictions. For example if it was in a royal forest they would find themselves rather limited in what they could do with their land. While there were cases where in principle you might be able to buy common land the people with rights of common over that land would maintain those rights.

You also get groups of non-aristocrats buying land. Town and cities could and did buy land as could monasteries (although they might expect gifts of land).

Sources:Norfolk Archaeology 1998 volume XLIII pages 79-98 and the City of Southampton Society book on Southampton Common (which covers a 13th century property dispute over Southampton Common).