Not only you can, but it is currently done: Guédelon
Guédelon Castle is the perfect example to prove you can do it. It's both an experimental archaeology project, and a touristic place.
The goal is to build a 13th-century medieval castle, using only medieval technique. even ressources are made with medieval technology, with wood coming from the forest and stone coming from a quarry.
Some of the knowledge is lost, but that's the goal of experimental technology: try to find the lost thing by doing it with the same constraints they had. The project started in 1997, and will probably end circa 2030.
Currently, seigniorial dweling and chapel is ended, while the walls are almost ended. Aside the castle, there is also a village with a wattermill, a smithy, and lot of other building for several crafters such as potters, ropemaker, carpenter... All of them work with medieval technique, at the begining with only theorical knowledge, and now with practical ones, as they experiment a lot.
This is a 30 year project, and by itself, proven to be possible. However, it is possible thanks to Michel Guyot, who bough several castles in order to restore them. Due to scientific, historical craft interest, lots of people work here for free. Also, lots of earnings come from both tourist and regional/governmental funds. Therefore, if you want to make a castle only for yourself, as a private project, it will be much more costly.