The Passuk in Ezra (6:3) says that when Koresh gave the Yidden permission to build the second Beis Hamikdash, he told them to make it 60 amos wide and 60 amos tall. The question arises, however, that the Mishnah (Middos 4:6) writes that the Heichal was 100 amos wide, long, and tall. The Rambam (Hilchos Beis Habechirah 4:3) writes that this talking about the Heichal built by the Bnei Hagolah, the Yidden who returned from the exile to Bavel and built the second Beis Hamikdash. (This is as opposed to the First Beis Hamikdash, or Hurdus's renovated second Beis Hamikdash, which were both 120 amos tall, as we will soon discuss.)
The simplest answer to this question would seem to be that Koresh did not know exactly how big it was supposed to be, so he said sixty, but he really did not care, and let them make it one hundred. However, the Medrash says (Esther Rabba 1:5) that Hashem was angry at Koresh for dividing the Beis Hamikdash in half, as during the first Beis Hamikdash it was 120 amos tall, and Koresh said to make it sixty. From this it sounds like Koresh says sixty specifically; and did not let more. One can say that when Daryavesh became king and gave the Yidden permission to continue building the Beis Hamikdash (after the construction was halted by Koresh and Achashverosh because of libels by the antisemites), he let them make it one hundred amos tall, however there is another Medrash (Bamidbar Rabba 14:18) which says that the second Beis Hamikdash was actually built sixty by sixty amos. [see Eitz Yosef on the Medrash there, who raises this contradiction between this and the Mishnah in Middos, but does not answer it.]
Furthermore, Josephus writes (Antiquities 15:11:1, Yosifun chapter 55) that Hurdus, in his speech to the Yidden to get them to let him build the Beis Hamikdash, says that the Yidden when they returned from Bavel, they did not build the Beis Hamikdash in its original size, but rather smaller, based on the instructions of the kings of Persia. From this we see that they actually made it smaller.
Now, about the width we could easily explain that it is talking about the interior, without the walls, and indeed from the Beis Horedes Hamayim in the south to the Mesibah in the north was sixty amos, if we don't include their exterior walls (Beis Horedes Hamayim-3, wall of the Ta-5, Ta-6, wall of the Heichal-6, the Heichal-20, wall of the Heichal-6, Ta-6, wall of the ta-5, Mesibah-3, total of sixty). We can alternatively say that it is talking about the length, since the Kodesh was forty amos long and the Kodesh Hakodoshim was twenty, this adds up to sixty.
However, the height is still an issue, as there is no measurements in the Mishnah about the Height that be added up to sixty and be explained that that is what the Passuk is referring to, and furthermore this has to be half as tall as Bayis Rishon.
The Ezras Kohanim (in one of his introductions, called Beis Hamiddos) tries to answer this question, but does not really come up with a good answer. Rebbi Shmuel Masnos, in his commentary on Ezra, says that each amah of Ezra was half of an amah of the one that Shlomo used, and since the Heichal in the first Beis Hamikdash was 30 amos tall (Melachim I 6:2), this is what Koresh gave permission to, but the passuk records it in the smaller amah to glorify the Beis Hamikdash,and says it was sixty amos. About the width, he says that if you add up the with of the Kodesh, which was twenty amos, and the width of the Ulam, which was ten amos in the First Beis Hamikdash (Melachim I 6:3), you come out with thirty, which if using the half amos is sixty. (It should be noted though, that width of the Kodesh is the measuremnt from north to south, and the width of the Ulam is from east to west, as that was the shorter measurement.) However, this works with the Passuk, but it doesn't explain the Medrashim.
To attempt to answer these questions, we first need to figure out the exact height of the Heichal in the First Beis Hamikdash. There are two Pesukim that mention the height, that seem to say two different measurements. One Passuk says that the kodesh was 30 amos tall (Melachim I 6:2), and one says that the Ulam was 120 amos tall (Divrei Hayamim II 3:4). The commentator on Divrei Hayamim (who is not Rashi, but a different Rishon) writes that the main floor was thirty amos tall, but there were upper stories on top, which added up to 120 amos. The Radak gives this answer, as well as another answer, that the Kodesh was actually only thirty amos tall, and only the Ulam had upper stories, which made the Ulam 120 amos tall.
Rashi on Ein Yaakov (Moed Katan 9a) writes that the 30 amos measurement includes even the amah of the klei orev (the spikes on the roof to prevent the birds from sitting there). [The regular Rashi printed in the Gemara in Moed Katan is actually not Rashi, rather it is based off Peirush Rabbeinu Gershom. The actual commentary of Rashi can be found partly in the Ein Yaakov and on the Rif, as well as in various manuscripts. It is printed in the margin of the Oz Vehadar Gemara under the title of Rashi Ksav Yad.] From this it comes out that Rashi explains the Passuk in Divrei Hayamim like the Radak; that the measurement of 120 is referring to the Ulam. We can also try to deduce this from the words of Rabbeinu Shmaya, a student of Rashi. In his Peirush on Maseches Middos he writes (Middos 4:7) that in the first Beis Hamikdash the Ulam was 120 amos high, as it says in Divrei Hayamim, even though the Kodesh was only thirty amos high.
[Based on this we can understand what Rashi says in a number of places, (see for example Yoma 51b) that in the first Beis Hamikdash the Kodesh was thirty amos tall, and in the second it was 100. The Tosfos Yom Tov, (Yoma 5:1) emends Rashi to say that in the second Beis Hamikdash the Kodesh was forty amos tall, as he seems to hold that Rashi holds like the commentator on Divrei Hayamim, that the whole building was 120 amos tall, possibly the Tosfos Yom tov held that this commentary is from Rashi himself. However, based on the above-mentioned Rashi in Moed Katan, there is no need to emend Rashi. Different Mefarshim (Chanukas Habayis Chefetz siman 55 and Lechem Shamayim Yoma 5:1) have already pointed out that Rashi can be explained this way.]
Now, there is another Passuk (Melachim I 6:20) which says that the Kodesh Hakodoshim was twenty amos tall. Based on Rashi there it comes out that while the Kodesh was thirty amos tall, the Kodesh Hakodoshim was only twenty amos tall. So, it comes out that there were three different heights: The Ulam was 120 amos, the Kodesh was thirty amos tall, and the Kodesh Hakodoshim was twenty amos tall.
Now, even though the Kodesh Hakodoshim and Kodesh were only twenty (or thirty) amos tall, it also had upper stories on top (which is included in the measurement, as even the roof spokes are included in the thirty amos of the Kodesh). This upper story on top of the Kodesh Hakodoshim is mentioned by Rashi, who explains (Melachim I 6:9-10) that there were two ceilings, with the lower one also being the floor of the attic. It is also mentioned with regards to Yoash, who was hidden in the "bedroom" (Melachim II 11:2) which the Midrash explains (see Rashi on the possuk) as referring to the attic of the Kodesh Hakodoshim. (As Rashi to Melachim I 6:20 explains, since the Kodesh Hakodoshim was shorter, the attic on top would have been on a lower level than the one on the Kodesh.)
On top of the Ulam, however, there would have been many attics, probably four, with each one being 30 amos tall (with its ceiling), adding up to a total of 120.
Now based on this we can explain the Passuk that says the structure of the second Beis Hamikdash was sixty amos, together with the Mishnah that says it was 100 amos tall. When Koresh gave the Yidden permission to build the Beis Hamikdash, he only gave them permission to build two floors, not four, (which is why Hashem was angry with him.) and since in the First Beis Hamikdash each floor was thirty amos tall, he said to build it sixty, and indeed this is how the Yidden started building it. However, Chagai Hanavi had a Nevuah (Chagai 2:9) that the Second Beis Hamikdash will be greater than the first. There is an argument in Gemara (Bava Basra 3a-b) in what way this is, with one opinion being that it stood for longer, 420 years as opposed to the First Beis Hamikdash’s 410, while another opinion is that it was taller, as the Kodesh in the First Beis Hamikdash was only thirty amos high, while here it was 100. Based on this Nevuah, the Yidden made each of the two floors they were able to make forty amos high, and together with the thickness of the ceilings it adds up to 100 amos. This was not going against the King's orders, however, as his main concern was that there should only be two floors, he only said the number sixty since in the first Beis Hamikdash that was the heights of two floors. However, the Midrash could still say that the First Beis Hamikdash was built sixty amos high, since when the started building it, that was the measurement they had in mind. (this also explains why the Midrash says that it was built (שנבנה) 60 amos tall, not that it was (שהיתה) 60 amos.
This is also what Hurdus meant when he said that the Yidden made it smaller because of the instructions of the king of Persia. Hurdus intended to build the Heichal back to its original height, by making the sloped roof 120 amos tall (this what it seems to be saying in Yosifun chapter 55). However, it seems like Hashem did not want this to happen, as this part of the roof collapsed (Antiquities 15:11:3). The Yidden got more wood and wanted to rebuild it, but before they had a chance to, the war that led to the churban broke out. (Wars of the Jews 5:1:5)
I have added sixteen pictures of the third Beis Hamikdash, as well as a new map of it.
My name is Mendel Lewis.
Hashem said to Yechezkel, "Its reading in the Torah is as great as its building. Go and say it to them, and they will occupy themselves to read the form of it in the Torah. And in reward for its reading, that they occupy themselves to read about it, I count it for them as if they were occupied with the building of it. (Tanchuma tzav 14)