Since the exact dimensions of the ark - Aron - are mentioned in the Torah (Shmos 25:10 and on), why didn't they make one to replace the missing one in the 2nd Temple?
Rashash (Yoma 53, in part, in my own translation):
It's difficult to understand why they didn't make another aron in the second temple. If it was because it's written "and put the testimony in the aron", implying that its main need is nothing but the testimony, isn't "and put the urim, etc., into the justice breastplate" also written about the breastplate? And nonetheless (according to Raavad whom I cited above at the end of chapter 1) they [=the urim, etc.] were absent in the second temple but they made the breastplate to complete the eight-garment [uniform].…
It may be because God repeated [Himself] in the command of putting the testimony into it (parashas T'ruma, verses 16 and 21), for He repeated [Himself] about it to [indicate that its absence] is indispensable.
He's referring to a general rule of d'rashos about kadashim: If a detail of an action is mentioned twice in the Torah, that detail is indispensable: without it, the action doesn't count. (His answer is already mentioned in another answer post here, but not in his name, so I figured it's worthwhile mentioning.)
Pardes Yosef (25:10, paragraph 1, in part, in my own translation):
I heard: Why was there no aron in the second temple? The great Rabbi Chayim Ozer of Vilna replied to the rabbi of Gur: For all the vessels, their work inaugurated them. In the second temple, there were no tablets. Therefore, there was no aron: because there was no [way] to inaugurate [it]. One needs to look into this.
He's referring to the Bavli (Sh'vuos 15 and elsewhere) which says that temple vessels were inaugurated when used (except the vessels that Moshe made).
Mikdash David (volume 1, 2:2, in part, in my own translation):
Behold! The Rambam ad loc. included only the form of the shulchan, m'nora, and golden mizbeach, but not the form of the aron. The Minchas Chinuch 95 stopped at this and inferred from this detail that the Rambam holds that it's impossible to make another aron: only the one Moshe made. And he wondered about [the Rambam]: Why couldn't they make another aron in the first temple, or in the future if another aron is revealed? See there.
It seems one can say: According to what we say in [Bavli] Sh'vuos, "for all the vessels Moshe made, their anointment inaugurated them; thereafter, their work inaugurated them", a status of [temple] work vessel is made by anointing or work. Now let's see — what work is done with the aron? We don't find work with the aron! If so, there is no work to sanctify it. (One cannot say that the sprinkling of blood atop the kapores should inaugurate it, because for that one doesn't need an aron at all….) If so, since there is no work to inaugurate the aron, it of necessity requires anointment to be made a work vessel. And anointment was only on the vessels Moshe made. So the aron must be the one Moshe made, only.
T'shuvos Chasam Sofer (YD 236, near the end, in my own translation):
This is why "I will meet with you there… from atop the kapores" is written [25:22]: to say that the reason for the aron and kapores is solely the purpose of meeting, to speak with Moshe from between the k'ruvim. When there's no prophet like Moshe, fit to meet with from between the k'ruvim, the aron is not indispensable.
My thanks to Rabbi Menachem Greenblatt of St. Louis, who pointed me to these sources.
Since the Aron was supposed to contain the luchos (whether one or both), a new one could not be made without them. Since the luchos were in the original aron, they were not available for the second temple. Had they attempted to make a new one, it would not have had any kedusha and just been an empty box. Note that the Menorah, Shulchan, and Mizbeach were made because they were able to be sanctified and used for the correct purpose.
“And into the Ark shall you place the Testimony that I will give you” (Sh’mos 25:21). This is the second time within the instructions for the Ark that G-d told Moshe to put “the Testimony” (referring to the Luchos, the stone tablets with the “Ten Commandments” carved into/through them; it also refers to the Torah, see http://rabbidmk.wordpress.com/2013/03/07/parashas-vayakhel-pekuday-5773/) into the Ark (see 25:16).
Please note below that the citation from Rabbeinu Bachye may be a typo and mean 25:18 but I have not had a chance to verify it.
which leads him to the conclusion
Others (e.g. Rosh and Tur) say that the verse means that it is forbidden to cover the Ark if the Luchos are not inside, or, put another way (see Rabbeinu Bachye towards the end of his commentary on 21:18) that it is forbidden for the Ark to not have the Luchos inside of it. (They are not all saying that this is what Rashi meant; Gur Aryeh makes this suggestion to explain the repetition, but assumes it is not what Rashi meant.) Based on this, many (e.g. Rashash on Yuma 53b and Chasam Sofer on our verses) explain that this is why there was no Ark in the Second Temple, as the Luchos were “hidden” (or in exile, see Yuma 53b) along with the Ark from the First Temple (shortly before its destruction). This is contrasted with the Choshen, the Kohain Gadol’s breastplate, which was worn in the Second Temple even though the “Urim v’Tumim” were also lost, despite the Torah also telling us (28:30) to put the Urim v’Tumim inside the Choshen. Since those instructions weren’t repeated, the Choshen was allowed (and needed for the Temple service) even without the Urim v’Tumim, while a Luchos-less Ark was not. Some (e.g. Meshech Chachmah) point out that there is a general rule regarding Temple service that whenever a law detail is repeated, the service is not valid without the detail being fulfilled; since putting the Luchos inside the Ark was repeated, the Ark cannot be used without them.
That the Ark isn’t really considered a valid “Ark” without the Luchos inside is explicitly stated by Ramban (40:2).
A very partial element of response. Following the Mishna Shekalim (6,2), the Ark was hidden and not missed. But following the Mishna Yoma (5, 2), the Ark was swept away.
The Gemoro Yoma (53B-54A) provides many details on the various opinions
According to many opinions, the Ark is not really missing but intentionally hidden in the Beith Hamikdash (two opinion about his place). So it is not surprising that they did not think to build another, if it was an unavoidable (despite a concern to protect it and its content) necessity to place an Ark, it would be enough to dig it up. I assume that it was prohibited to dig it up, perhaps because they new that the second Bet Hamikdash would be destroyed. So, for the same reason, they did not build a new Aron. For the opinion that the Aron was in Bavel, I don't know. See also the Ben Yehoyada about this Machloket.