Answer: The Gemara (Sukka 3a) writes that we are not required to attach a mezuza to a house which is smaller than four amos by four amos. According to R’ Chaim Naeh (Shiurei Torah 13:25) this measures up to just under 3.7m2. Even the Chazon Ish (quoted in Shiurin Shel Torah 10:8), who maintains that an amah is larger, holds that one should follow the smaller size in this case.
While the Rosh (Mezuza 16) writes that both the width and length of the room need to be at least four amos, Rambam (Mezuza 6:2) and Shulchan Aruch (YD 286:13) write that it depends only on the size of the area. Thus, the Shach (YD 286:23) and Aruch Hashulchan (YD 286:21) hold that if either the length or width of the room is less than four amos, one should affix a mezuza without reciting a beracha.
The Pischei Teshuva (YD 286:11) quotes the Chamudei Daniel who writes that one is obligated to affix a mezuza to a room that is less than 4 amos by 4 amos, if it serves a proper purpose (See Minchas Yitzchak 3:103; 4:92; Yechave Daas 4:51; Ohr Letzion 1:14). Nonetheless, R’ Ovadia Yosef quotes R’ Yosef Chaim Sonnenfeld (Salmas Chaim YD:105) and others who write that we do not follow the Chamudei Daniel (See Igros Moshe, end YD 1:181).
The Shulchan Aruch (YD 286:2) writes that utility and storage rooms require a mezuza. A changing room, however, is exempt, as it is considered disrespectful to affix a mezuza there. The Bach (YD 286:5) quotes the Kol Bo who writes that one is exempt from affixing a mezuza onto a laundry room, due to the unpleasant odours there. However, R’ Moshe Sternbuch (Teshuvos Vehanhagos 2:548) explains that this doesn’t apply to modern day laundry rooms.
Nonetheless, R’ Shmuel Wosner (Shevet Halevi 2:152) writes that if the laundry room contains soiled clothes and is less than 4 amos by 4 amos it doesn’t require a mezuza. Elsewhere, (Shevet Halevi 5:167; 7:171), R’ Wosner notes that some place a mezuza from the laundry room going out into the adjacent room.
In conclusion, one does not need to affix a mezuza to a small laundry room.