The Shulchan Aruch rules that a storage room requires a mezuzah, and this is accepted practice. In a normal case, the custom is even to make a berachah on the placement.
Even though the owner or renter does not “live” there, it is considered to be an extension of his habitat as long as he enters it from time to time to tend to his items. For example, a barn is obligated, not because it houses cows, but rather because the owner enters there to tend his cows. 
The question then arises: How often does an owner or renter need to visit his storage room in order to consider it obligated in mezuzah?
R’ Eliashiv is reported to have required a mezuzah only if one enters it frequently.  R’ Moshe Heinemann is quoted as ruling that a storage room needs a mezuzah only if it is accessed at least once every thirty days. 
Your friend’s rabbi obviously follows one of these opinions, and thus he ruled that a self-storage room that is visited rarely does not need a mezuzah. He may also have ruled leniently because it is a rental, and its mezuzah obligation is Rabbinic 
In contrast, R’ Moshe Feinstein writes that even if a person enters his storage room infrequently, it needs a mezuzah,  and many people follow this custom and affix a mezuzah without a berachah.
One important point: You should only place a mezuzah there if you are confident that it will not be stolen or disgraced during your long absence.
 Shulchan Aruch Y.D. 286:2. Agur B’ohalecha (34:2:4) points out that even though Rambam exempts storage rooms, the minhag follows this ruling even to the point of making a berachah. See Aruch HaShulchan 286:6.
 Shach 286:2, citing Derishah; Noda Bi-Yehuda O.C. 2:47.
 Oral ruling, cited in Agur B’ohalecha 34:19:39 and Avnei Yoshfeh 1:209:2.
 R. Mordechai Frankel, “Insights from the Institute”, Kashrus Kurrents, Autumn 2008 (Star-K online http://tinyurl.com/nhnjrw4).
 Pitchei Teshuvah 286:16, citing R. Akiva Eiger. See discussion in Agur B’ohalecha 30:3:7.
 Igros Moshe Y.D. 2:141:2.
 Shach 286:7.