GEMARA

Beraisa: If a house does not have four by four Amos it is exempt from Mezuzah and Ma'akah (a protective wall around the roof), it cannot receive Tzara'as...

3b - Suggestion: This is like Rebbi (who says that a Sukah must be four by four Amos). It is unlike Chachamim.

Rejection: It is even like Chachamim. Chachamim allow a smaller Sukah because it is Diras Arai. Chachamim agree that to be called a 'Bayis', which connotes Diras Keva, it must be four by four. People do not live in something smaller.

Regarding all of these (Mezuzah, Ma'akah, Tzara'as...) it says "Bayis".

RISHONIM

Rif (Hilchos Mezuzah 6b (printed after Menachos)): The Gemara exempts a house from a Mezuzah if it does not have four by four Amos.

Rambam (Hilchos Mezuzah 6:2): If a house does not contain four by four Amos it is exempt from a Mezuzah. If it is big enough Lerabe'a (to make a square) four Amos by four Amos evenly it is obligated, even if it is round or pentagonal and all the more so if it was longer than it is wide.

Rosh (on Rif, and Hilchos Mezuzah Siman 16): All texts of the Gemara (Sukah 3a) say that a house must be four Amos. The Rif changes it to say four by four Amos, for less than this is not fit for Dirah. This is unlike the Rambam, who says that if it was longer than it is wide and it is big enough Lerabe'a four by four Amos it is obligated. Every matter mentioned there (Sukah 3a) requires four by four Amos.

Note: Our texts say like the Rif.

Question (Sho'el in Teshuvas Chasam Sofer YD 280): The Mishnah (Ohalos 3:7) requires an Ohel for Tum'as Mes to be 'a Tefach by a Tefach and a Tefach high, Meruba.' The Bartenura explains that it says 'Meruba' (square) to exclude a half Tefach by two Tefachim. This shows that otherwise such a rectangle is considered 'a Tefach by a Tefach'!

Answer (Chasam Sofer, ibid.): The Rosh can explain that 'Meruba' is needed for the height. Further, all can agree that 'four by four' connotes a square. Regarding Mezuzah it says 'there is (Kedei Lerabe'a)', therefore the Rambam holds that that it suffices to contain the area. The SMA says that we are stringent to obligate a Mezuzah in order that (awareness of) Hash-m will be in front of a person. This is good reasoning, but it is not necessary.

Ri'az (in Shiltei ha'Giborim 2): If a big house is open to a small house less than four by four Amos, whether the door hinge is on the side of the big or small house, the small house requires a Mezuzah. This is because it is like an opening of the big house.

Rambam (Hilchos Sukah 4:7): A round Sukah is Kosher if it is big enough Lerabe'a seven Tefachim by seven Tefachim inside, even though it has no corners.

POSKIM

Shulchan Aruch (YD 286:13): If a house does not contain four by four Amos it is exempt from a Mezuzah. If it is big enough Lerabe'a four by four Amos evenly it is obligated, even if it is longer than it is wide or it is round or pentagonal.

Tur: The Rambam obligates a Mezuzah as long as the area of the house is no less than that of a square four by four Amos. The Rosh argues. He requires that the house hold a square four Amos by four Amos.

Bach (OC 634 DH u'Mah she'Chosav): The Rambam explains regarding Mezuzah that a house is exempt unless a square four Amos by four Amos fits inside. However, the Rosh argues and says that a room with the area of a square four by four Amos needs a Mezuzah.

Note: The Rosh explicitly says that he argues with the Rambam. However, it is possible to explain the Rambam either way (Divrei Chamudos on the Rosh (50).) Here in Hilchos Mezuzah the Bach does not say that he argues with the Tur, even though the Tur explicitly says that the Rosh is more lenient than the Rambam and does not obligate unless there is (room for) a square four by four Amos.

Beis Yosef (DH u'Mah she'Chosav): The Halachah follows the Rambam, for only the Rosh argues with him. 'Four by four Amos' does not connote that it holds such a square. It suffices if it holds the area of such a square. Such a house is fitting for Dirah.

Question (Taz OC 634:2): How did the Rosh, Tur and Beis Yosef understand that regarding Mezuzah the Rambam does not require that the length and width of the house are both at least four Amos? Why should it be different than (the seven by seven Tefachim needed for) a Sukah? The Rambam says '...Lerabe'a four Amos by four Amos evenly!' Rather, he requires that it can hold such a square, just like he requires that a Sukah can hold a square of seven by seven. The Rosh says that he argues with the Rambam because he misunderstood the Rambam. Really, they agree.

Note: The Bach (ibid.) concludes that also regarding Sukah the Rambam holds that it suffices if its area equals that of a square seven by seven. Alternatively, perhaps a tradition from Sinai requires that a Sukah contain a square of the required Shi'ur:

Shach (23): Also R. Yerucham holds like the Rosh. Therefore, one should not bless on a Mezuzah in a house that has the proper area but is less than four Amos wide. Alternatively, he should bless on a house that is four wide and intend to exempt this as well.

Pischei Teshuvah (11): Four Amos is needed only for a house. A Beis Sha'ar, balcony or garden need not be four by four. It seems that if in a big house there are small rooms for storing things they require a Mezuzah.

Kitzur Shulchan Aruch (11:10): If a house does not holds a square of four by four Amos but it contains the area of such a square, some say that it is obligated in Mezuzah and some exempt.

Note: Surely, there must be a minimal width. A house a Tefach wide is surely not fit for Dirah! The Taz (634:1) says similarly regarding Sukah. He concludes that a Sukah must contain a square seven by seven, for otherwise Chachamim should have taught the minimum width!