Moshe, this is a very interesting question!

I found that the Chazon Ish (Orach Chayim, Eruvin, Likutim 110(2):20, DH v'Amnam Im, page 322) writes that the wall must both protect the city and also ensure that the city is not exposed. It seems that there is no specific height for the wall; the crucial point is that it must defend the city against potential enemies.

I will attemmpt to elucidate what the sources are for this explanantion.

1. The Chazon Ish cites a Gemara in Megilah (5b) that relates that Chizkiyah read Megilas Esther in Teveryah (Tiberias) on both the 14th and 15th if Adar because he was in doubt about whether it is considered a walled city (for which the Halachah is that one reads the Megilah on the 15th). The reason for Chizkiyah's doubt was that one side of Teveryah is the Sea of Kineret. The Gemara questions Chizkiyah from a Beraisa which derives from Vayikra 25:31 that Batei ha'Chatzerim are not considered houses in a walled city because they do not possess a wall "Saviv" -- "around" them, which teaches that Teveryah is not considered a walled city because the sea is its wall. What, then, was Chizkiyah's doubt? To answer this challenge from the Beraisa, the Gemara distinguishes between the definition of a wall required for the laws of Megilah and the definition of the wall required for the laws of Batei Arei Chomah. Since the word "Chomah" is not stated explicitly concerning Purim (see Rashi), it is possible that Teveryah is considered a "fortified" city because the sea on one side protects it, and thus it may be defined as a "walled" city for Megilah purposes. In contrast, with regard to houses in a walled city (where one has only one year to redeem a house that he sold), it is clear that Teveryah is not considered a walled city because it is exposed on its sea side, even though it is defensible.

2. The Chazon Ish writes that certainly Arei Chomah must possess a proper wall, while a "Mesipas" (a low partition made of pieces of wood) is not sufficient. For the laws of Batei Arei Chomah, the city must be both defensible and also not exposed. The Chazon Ish cites a source for this from Devarim 3:5, "Cities fortified with a high wall."

3. The above verse is also cited by Rashi in Erchin (33b, DH v'Ne'emar) as a source for the Gemara's statement there that "Chomah" is essential for the status of a walled city. We learn from this that the wall should be suitable for a fortress and capable of defending the city from invading armies.

Kol Tuv,

Dovid Bloom

I found further sources for the distinction between what is the definition of a house in Arei Chomah and between the wall required in a refuge city or the wall required to obligate the city to read the Megilah on the 15th of Adar.

1. The Birkei Yosef (OC 688:4) writes that the custom in the holy city of Hebron is to read the Megilah both on the 14th and the 15th of Adar, because there is a doubt that Hebron might be a walled city from the days of Yehoshua bin Nun.

2. The question is that in Sefer Yehoshua (20:7) and the Gemara in Makos (9b) we are told that Yehoshua set aside Kiryat Arba, which is Hebron, as an Ir Miklat, a refuge city. How can this be reconciled with the Gemara in Erchin (33b) which states that

the Arei Miklat must be only medium-sized cities (because if the cities are too large, we are concerned that the Goel ha'Dam might find his way into the large center and take revenge for his relative who was inadvertently killed by the Rotze'ach who now finds shelter in the Ir Miklat), which, as Rashi explains, means that the cities do not possess a wall. Since Hebron is an Ir Miklat, how is it possible that there is a doubt that the Megilah should be read there on the 15th? An Ir Miklat does not have a wall!

3. To answer this question, I would like to suggest a Chidush and assert that the refuge cities and the Levite cities possessed a "Kir" -- a smaller wall -- but not a "Chomah." My source for this is Bamidbar 35:4, where the Torah states that the Levite cities possessed a "Kir." Onkelos translates this as "Kotel." While the Levite cities have a "Kir" or partition, in contrast the verse in Devarim (3:5) mentions "fortified cities with a high Chomah." This suggests that Arei Chomah are fortresses. A Chomah is a much more substantial wall than a Kir.

4. Now we may understand a little better what Rashi means in Erchin (33b, DH Ela) when he writes that the refuge cities should be medium-sized cities which do not have a wall. When Rashi writes that it has no Chomah, he does not mean that it has no wall at all, but rather that its wall is not similar to that of a fortified city. This is also what the Gemara means (three lines from the bottom on 33b): The word "Chomah" is used specifically in connection with Arei Chomah, which is why Rashi cites the verse that the cities were fortresses; "Chomah" means a walled fortress, in contrast with other cities -- such as refuge cities or "walled" cities which do not possess a Chomah, but only a Kir.

5. Now we can understand why the Megilah is read on the 15th of Adar in Hebron even though Hebron does not possess a Chomah. In my previous answer, I quoted Rashi in Megilah (5b DH Ki) who writes that the Book of Esther does not state explicitly that to read on the 15th the city must possess a Chomah. Rather, Megilas Esther distinguishes between Perazim (open cities) and non-Perazim (cities which are not open). To read the Megilah on the 15th, the city mut be closed -- but it need not be a fortress. Therefore, the Megilah is read on the 15th in Hebron because it has a wall -- a Kir -- even though it does not possess a Chomah and is not a fortress.

(I saw similar ideas to the above in Sefer Erech Dal on Maseches Erchin (this Sefer has a Haskamah from the Maharal Diskin). He writes (on 33b, DH Amnam) that Arei Chomah must be fortified cities which contain places which are ready for wartime and weapon storehouses ready for immediate use. The fortress is continually guarded by soldiers. He writes that Hebron possessed a medium-sized wall, not a fortress.)

Kol Tuv,

Dovid Bloom

Moshe, thank you for showing me this Derech Emunah.

The conclusion of the Derech Emunah seems to lean towards our conclusion -- that the wall of an Ir Chomah must be a strong, fortified wall. However, apart from not citing the Chazon Ish that we found, he also does not mention the Gemara in Erchin (end of 33b) which points out that "Chomah" is mentioned explicitly in connection with Arei Chomah, which Rashi writes is learned from Devarim 3:5: "All these are fortified cities with a high wall and locked gate." Tosfos in Bava Metzia (53b, DH d'Naful) also is consistent with our argument, because Tosfos writes that in order for one to be allowed to eat Ma'aser Sheni, it suffices that the majority of the wall be intact. This is possible only for Ma'aser Sheni, because the Torah does not state explicitly "Chomah" for Ma'aser Sheni, but it is not a sufficient wall for Ir Chomah because the latter must be a fortress.

Kol Tuv,

Dovid Bloom