The Halachic source for the requirement:

Interestingly, the accustomed obligation to have a Mechitza in Shul or by other events is not explicitly recorded in the Rambam or Shulchan Aruch, and the later Poskim[1] collect the various Talmudic and Halachic sources which lead to its requirement.  

1. Source #1 from Scripture-Separation by Eulogy: The verse in Zecharia[2] states regarding the future era, “And the land will eulogize [the death of a certain individual[3]]. Each family will eulogize him on their own. The family of the household of David will eulogize him alone with the female members of the family eulogizing him on their own. The family of the household of Nathan will eulogize him on their own, with the female members of the family eulogizing him on their own.” In other words, Scripture is stating that even in the future era during the time of a eulogy, the men and women will be separated from each other.[4] The Talmud[5] explains that the sages learned from this verse that if even in the future era when there will no longer be an evil inclination, a separation of men and women is required even by a eulogy, which is a sad event that does not arouse the evil inclination, all the more so is a separation required now when the evil inclination exists, and all the more so by a happy and joyous event. It was the expounding of this verse which served as the clause which permitted the sages to renovate the first Temple and build a womens section, as will be explained next. [From this source, the Poskim learn that the requirement of separation men and women applies not just by a Shul, but by other events as well.]
2. Source #2 from Talmud and Poskim-Women’s balcony by Simchas Beis Hashoeiva:[6] The Mishneh[7] states that on the leave of the first day of the holiday of Sukkos they descended to the women’s quarters of the temple [i.e. Ezras Nashim] and made there a great Tikkun, productive correction. The Talmud[8] asks as to what great correction was made, and explains that they would build a temporary women’s section in the temple courtyard in order to separate the men and women from each other during the Simchas Beis Hashoeiva festivities. The men were positioned on the ground of the Ezras Nashim while the women were positioned on top, on a balcony that was specially built for this occasion. The balcony protruded from the inner walls of the temple which surrounded the Ezras Nashim, and was built elevated on the upper part of the wall. The allowance to renovate the temple [which was built under the direct guidance of G-d through his prophets Gad and Nathan], was learned from the above verse in Scripture which requires separation of men and women even during times of eulogy and even in the future era. Originally, prior to the renovation of this balcony, during the Simchas Beis Hashoeiva festivities, the men were outside the temple, on the Temple Mount, while the women were inside the Ezras Nashim section of the temple. However, this led to lightheadedness and frivolity and hence they changed the order that the men should be in the inside and the women on the outside. However, this too led to lightheadedness and frivolity which eventually led to the renovation of the balconies for the women to sit on top and be separate from the men who are on bottom. [From this source, the Poskim learn that the requirement of separation men and women by a Shul, which is similar to a Temple.]
3. Source #3 from Talmud And Poskim-Event Dividers: The Talmud[9] states that both Abayey and Rava would create a division between men and women by events in which men and women gather [such as by a Torah lesson or wedding[10]]. Abayey would organize a series of earthenware jugs between the men and women, which would prevent them from crossing into the other genders designated area, as this would cause the jugs to shake against each other and make a loud sound. Rava would organize a series of bamboo sticks between the men and women which would make noise in the event that someone tried to cross past them. Avin had cursed the festivals in which men and women would gather together by events, such as by Torah lessons, [as these gatherings would lead to intermingling between the men and women[11], and also lead to them looking at each other in a sinful way[12]]. This concept of setting up a divider between men and women by Torah events is recorded in the Poskim.[13]
4. Source #4 from Poskim-Festival patrols: Based on the above Talmudic teaching in #2[14], and based on a further Talmudic dictum[15], the Poskim[16] rule that the Jewish courts are obligated to appoint police to patrol during the festivals throughout the gardens and orchards and rivers to make sure that men and women are not gathered together to eat and drink and come to sin. Likewise, they are to warn the nation to not have intermingling between men and women even in their own homes during the festive meal, as they may be drawn to consume alcohol and be drawn towards sin. The Achronim[17] conclude that in today’s times, this patrol is required even during regular times of year, as due to our great sins, intermingling of men and women has become customary even during other times of the year. [From this source, the Poskim learn that the requirement of separation men and women applies not just by a Shul, but by other events as well.]
5. Source #5 from Poskim-Sheva Brachos divisions: The Poskim[18] rule that it is forbidden for the blessing of Shehasimcha Bemiono to be recited by the festive meal of a wedding [i.e. Sheva Brachos] if the men and women are sitting in the same room and are able to see each other. The reason for this is because there is no joy above in heaven in a place that the evil inclination has control and in which sin is found, and in all areas in which men and women are found together there is worry of forbidden thought. Some Poskim[19], however, record that the custom is to no longer be careful in this and to recite Shehasimcha Bemiono even if the meal takes place with men and women together in the same room. The suggested reason for this is because in today’s times men are very accustomed to being around women and hence are not led to sinful thoughts as a result of seeing them. [From this source, the Poskim debate if the requirement of separation between men and women applies by events, such as a wedding.]
6. Source for lack of need of Mechitza in home meal: The law is that it is forbidden to have a Mechitza present between the group of people who would eat the Pesach sacrifice on the night of Pesach.[20] Now, being that women are also obligated in the Mitzvah, and were prohibited from making their own group for eating, from here it is proven that it is permitted for men and women to eat a meal together without a Mechitza.[21] Nonetheless, there are meals in which a Mechitza is required due to it being a public event, as will be explained.

[1] See Chasam Sofer C.M. 190; Maharam Shick 77; Machaneh Chaim 3:10; Beis Hillel 104; Seridei Eish O.C. 14; Igros Moshe O.C. 1:39, 41, 42; 2:40; 109; 3:23-24; Divrei Yoel O.C. 10; Likkutei Minchas Yitzchak 14; Tzitz Eliezer 7:8; Mishneh Halachos 7:12

[2] 12:12

[3] The Talmud Sukkah 52a records a dispute as to whose death is being eulogized, with some saying that they are eulogizing Moshiach Ben Yosef, who will be killed during the war of Gog Umagiog. Others say that they are eulogizing the evil inclination, who will be slaughtered in the future.

[4] Rashi Sukkah 51b

[5] Sukkah 51b

[6] See Chasam Sofer C.M. Hashmatos 190; Igros Moshe O.C. 1:39 for his explanation and understanding behind this Gemara

[7] Sukkah 51a

[8] Sukkah 51b; Rambam Mishneh Torah Lulav 8:12

[9] Kiddushin 81a

[10] Rashi ibid

[11] Rashi ibid

[12] Tosafus ibid

[13] Admur 315:3 “Such as a Divider that is made at the time of the Drasha to divide between men and women”; Rama 315:1; Taz 315:1; Mordechai Shabbos 3 Remez 311; M”B 315:5

[14] Maggid Mishneh 6:21

[15] Avin in Kiddushin 81a “Sakva Deshasa Rigla”

[16] Admur 529:13; Michaber 529:4; Tur 429 and 529; Rambam Hilchos Yom Tov 6:21; Avin in Kiddushin 81a “Sakva Deshasa Rigla” as explains Rashi ibid “May the festivals which are opportune for sin die, as men and women gather together to hear the lecture and end up intermingling” and Tosafus ibid “they look at each other”

[17] M”B 529:22; Shaar Hatziyon 529:21

[18] Beis Shmuel E.H. 62:11; Bach E.H. 62:12; Beir Heiytiv 62:11; Sefer Haminhagim Tirana Hagahos 14; Sefer Chassidim 393, brought in Levush O.C. Vol. 2 [Levush Hachor] Minhagim 36, “When man and woman are together and see each other such as by meal of a wedding, then one does not say the blessing of Shehasimcha Bimiono, as there is no joy in front of G-d when sinful thoughts are being pondered”; Orchos Tzadikim Erech Simcha; Kitzur SHU”A 149:1

[19] Levush O.C. Vol. 2 [Levush Hachor] Minhagim 36 [end of Sefer, last Halacha], brought in Pischeiy Teshuvah E.H. 62:18, after quoting the Sefer Chassidim ibid he concludes “today we are no longer accustomed to be careful in this, and possibly the reason for this is because today’s women are much more accustomed to be amongst men, and hence there is not as much sinful thought caused by this, as we consider them like a “white geese”[see Brachos 20b] due to having become accustomed to then being amongst us, and since they have become used to it, they have become used to it and may G-d bless us to see the joy of the coming of our Redeemer and the building of our holy temple.”

[20] See Pesachim 64b; 86a; 91a

[21] Igros Moshe O.C. 1:41