You honestly would not own very much, and almost all of it would have been hand made or handed down. Peasants generally lived in very simple wooden huts and the furniture they did have was very simple too. Chairs as we recognize them today were for the rich, the poor sat on the ground or on wooden stools or benches. There might have been a wooden chest for storage which could double as a table or there might have been a separate wooden table. Beds were piles of hay covered in blankets or stuffed into sacks, often with a log as a headrest.

It was a very difficult, uncomfortable life. Peasants couldn't really afford to buy anything brand new, so they had to be self-sufficient. While you may not have been a carpenter by trade, you knew enough to fashion basic furnishings or knew one who might supply you with what you needed. Once you had that piece of furniture, you took care of it as best you could and did your best to pass it down to your descendents.

Lucy Worsley wrote a very encompassing easy-to-read book (and tv series) about the home (and its furnishings) called If Walls Could Talk. The earliest parts of each chapter/episode gives a good idea of life in the Middle Ages.

Edit: I forgot to add that all four episodes of the show are on Youtube under the name "History of the Home."