The Sages determined that a room smaller than four by four amot does not meet the minimal criteria of a dwelling suitable for human habitation, and it is thus exempt from many halachic considerations including mezuzah obligation.
The room need not be a perfect square. Even if it is rectangular or even circular, as long as it is at least four amot in its length and four amot in its width, it is obligated in mezuzah.
If the kitchen is sixteen amot sq. in area, but one of its dimensions is less than four amot (e.g. it is two amot wide and eight amot long), you should still affix a mezuzah. However, you should then not make a berachah since it is not four by four amot in both length and width. The mezuzah should be placed on the right side of one going into the kitchen.
If you are talking about a real “Manhattan kitchen,” meaning that its total area is less than 16 amot sq., even according to the minimal calculation, many authorities exempt it from mezuzah obligation altogether.
However, others point out that though the kitchen itself is exempt, its door deserves a mezuzah on its right side leaving the kitchen as it also serves as an entrance into the dining room. In deference to this opinion, it is common practice to affix a mezuzah on the right side of one going from the tiny kitchen (without a blessing).
 Sukkah 3a.
 Chovas HaDar 4:7.
 Shulchan Aruch Y.D. 286:13; Shach Y.D. 286:23; cf. Kuntres HaMezuzah 286:149. If the width is less than seven Tefachim, its obligation is extremely doubtful.
 Agur B’ohalecha 19:7:21; Cf. Shevet HaLevi 2:152.
 Mezuzot Beitecha 286:52, citing Chazon Ish; Agur B’ohalecha 18:28:84; Cf. Pitchei Teshuvah 286:11, citing Chamudei Daniel.
 Chazon Ish Y.D. 168:3; Shevet HaLevi 2:152; Teshuvos V’Hanhagos 1:653; Agur B’ohalecha 19:6.