Gemara

(Mishnah): A Reshus ha'Rabim is 16 Amos wide.

Shabbos 6a (Beraisa #1): Reshus ha'Rabim is a road between cities, a big market, and alleyways in a city that are open on both ends. This is a bona fide Reshus ha'Rabim.

Question: Why didn't the Tana include the Midbar?

(Beraisa #2): Reshus ha'Rabim is a road between cities, a big market, open alleyways and the Midbar.

Answer (Abaye): One Beraisa discusses when Benei Yisrael were in the Midbar. The other Beraisa discusses nowadays.

7a: Rav Dimi holds that between the pillars (in the market) is a Karmelis;

Version #1: R. Zeira said that the platform is a Karmelis, for it is inconvenient to walk there. He would hold that between the pillars, which is convenient, is not Karmelis (rather, it is Reshus ha'Rabim).

Version #2: Between the pillars is Reshus ha'Rabim, because sometimes people walk there.

98a (Rav): One who carries four Amos in a Reshus ha'Rabim with a ceiling is exempt, for this is not like the encampment in the Midbar.

Eruvin 6b (Rabah bar bar Chanah): Had they not locked the doors of Yerushalayim at night, it would have been Reshus ha'Rabim. One who carried in it would have been liable;

(Ula): If they didn't lock the doors of the gates of Mechuza at night, it would be Reshus ha'Rabim.

Rishonim

The Rif (Shabbos 2a and Eruvin 2a) brings Beraisa #1 and the teaching about Yerushalayim.

Rambam (Hilchos Shabbos 14:1): Reshus ha'Rabim is a Midbar, forest, market, or a road open to them.

Question (Magid Mishneh): The Gemara says that the Midbar was a Reshus ha'Rabim only when Yisrael were there!

Answer (Kesef Mishneh): The Rambam's son explains that when Yisrael had an arranged encampment in the Midbar, it was like a valley or field. Nowadays, anyone may go anywhere they want, so it is Reshus ha'Rabim. A forest is a Reshus ha'Rabim because everyone needs to go there for wood, and it is easy to walk there.

Rambam (ibid): The road must be 16 Amos wide, without a roof.

Magid Mishneh" The Rashba says that if a Mavuy is open to the length of Reshus ha'Rabim, it is Reshus ha'Rabim even if it is not 16 Amos wide, but not if it is open to the width of Reshus ha'Rabim. There is no proof for this from the Gemara. Some Ge'onim and Acharonim say that Reshus ha'Rabim must have 600,000, like the encampment in the Midbar. There is no hint for this from the Gemara. It is not primary. The Ramban and Rashba say so.

Rosh (Eruvin 1:8): Rashi says that Reshus is 16 Amos wide in a city with 600,000 and there is no wall, or the Reshus ha'Rabim went from gate to gate, so it is open on two sides, like the the encampment in the Midbar.

Hagahos Ashri: Shabbos 6b supports this. It says that when Yisrael dwelled in the Midbar, it was a Reshus ha'Rabim.

Rosh (ibid): The Ri says that if Mavo'os 13 and a third Amos wide are open on two sides to a Reshus ha'Rabim 16 Amos wide, since many go through constantly, it is Reshus ha'Rabim mid'Oraisa. He proves this from Shabbos 7a, in which we say that Reshus ha'Rabim between the pillars is Reshus ha'Rabim, even though the platform and pillars diminish (the width from 16 Amos). Even the opinion that calls it a Karmelis would agree that that it is Reshus ha'Rabim when people can walk through easily.

Poskim

Shulchan Aruch (OC 303:18): Some say that women are lenient to go out with jewelry, for nowadays we do not have Reshus ha'Rabim. Our Reshus ha'Rabim is Karmelis.

Shulchan Aruch (OC 345:7): Reshus ha'Rabim is a street or market 16 Amos wide without a roof and without a wall. Even if it has a wall, if it is open from one gate to the other (Rema - and the doors are not locked at night), it is a Reshus ha'Rabim.

Magen Avraham (5): Also an intercity road is a Reshus ha'Rabim.

Taz (5): It seems that if one door is locked, it is not Reshus ha'Rabim, for it is not Mefulash (open on two sides).

Rebuttal (Kaf ha'Chayim 34): The Magid Mishneh says so in the name of the Rashba, and Tosfos agrees, but the Shulchan Aruch (364:2) rules like the Rif, Rosh and Rashi, that one door is not a full Mechitzah, therefore, a door is needed also on the fourth side.

Mishnah Berurah (22): The Rema connotes that even if they could be locked, but they are not, it is Reshus ha'Rabim. This is like the first opinion below (364:2).

Kaf ha'Chayim (35): The Rashba says that the city square is a Reshus ha'Rabim even if the doors are locked. The Birkei Yosef says that the primary opinion says that it is not.

Shulchan Aruch (ibid): Some say that if 600,000 do not pass through every day, it is not Reshus ha'Rabim.

Beis Yosef (DH uv'Reish): The Rosh cited Rashi without dissent. This connotes that he agrees that 600,000 are required.

Note: Immediately afterwards, the Rosh cites the Ri, who says only that many must pass through!

Beis Yosef (ibid.): Also in Beitzah (3:2) the Rosh said that nowadays it is not common to find Reshus ha'Rabim in which 600,000 pass every day. Also Semag, Semak and Tosfos say so. However, the Rambam never mentions this condition. The Ran says that if it were true, the Gemara would have mentioned it!

Taz (6): Above, the Tur challenged Sefer ha'Terumos, because nowadays we do not have Reshus ha'Rabim (for cities do not have 600,000). The Beis Yosef said that also the Magid Mishneh, citing the Rashba, asked this. The Magid Mishneh and Rashba do not require 600,000! The Maharshal and Mas'as Binyamin say that one must be stringent like the first opinion. The Rema (brought below) says that we do not have Reshus ha'Rabim, and people say this. This is the majority opinion. One may be stringent, but he cannot protest against those who are lenient.

Magen Avraham (7): Above (303:18) and elsewhere, the Shulchan Aruch is lenient. Most Poskim are lenient.

Rebuttal (Bi'ur Halachah DH she'Ein): Many rejected this proof. In Siman 303, the Shulchan Aruch merely said how some try to defend the practice! In many places the Shulchan Aruch forbids (Stam) matters forbidden only in Reshus ha'Rabim, e.g. for a tailor to go out with a needle shortly before dark. Chachamim decreed not to fulfill Shofar on Shabbos. Surely, this was due to a practical concern for Reshus ha'Rabim! The Ramban says that even Rashi requires 600,000 only in a city, but not for intercity roads. One may rely on the lenient opinion only if there is another reason to be lenient.

Mishnah Berurah (23): Even though in the Midbar there were also women, children and Eruv Rav, we follow the number explicit in the Torah. It seems that the Shulchan Aruch favors the stringent opinion, for it calls the lenient opinion 'some say.' We cannot protest against one who is lenient; a Ba'al Nefesh should be stringent.

Kaf ha'Chayim (39): Eshel Avraham (8) says that seemingly, the lenient opinion should require 600,000 Yisre'elim, but the Shulchan Aruch and Poskim connote that Nochrim count.