Gemara

7b (Mishnah - R. Eliezer): Any woman who did not see blood for three periods is Dayah Sha'atah.

9b (Beraisa - R. Eliezer): A case occurred, a girl did not see blood for three periods, Chachamim ruled that she is Dayah Sha'atah.

Chachamim: We do not bring proofs from She'as ha'Dechak (pressed circumstances)!

Question: Why was it considered She'as ha'Dechak?

Answer #1: There was a famine.

Answer #2: She had touched much Taharos, so Chachamim were reluctant to cause a big loss.

6a (Beraisa): A case occurred, and Rebbi ruled like R. Eliezer. After he remembered he said "in She'as ha'Dechak, one may rely on R. Eliezer's opinion."

Suggestion: "After he remembered" means that he remembered that the Halachah does not follow R. Eliezer, rather, Chachamim.

Rejection: If so, even in She'as ha'Dechak, he could not rely on R. Eliezer's opinion!

Answer: The Halachah was not decided like either of them. He remembered that the majority argue with him, so l'Chatchilah one should rule like Chachamim, but in She'as ha'Dechak one may rely on R. Eliezer.

Bava Kama 117a (Beraisa): At first they used to obligate for being Metamei another's Peros, or pouring wine to idolatry. Later, they included Medame'a (mixing Chulin with Terumah, which forbids it to a non-Kohen).

At first they fined only when a major loss resulted. Later, they fined even for a small loss.

Avodah Zarah 35b (Rav Yosef): Rebbi was once in a place where there was not enough bread for the Talmidim.

Rebbi: Is there no baker here?!

People understood from this that Rebbi permits bread of a Nochri baker;

This was wrong. Rebbi referred to a Yisrael baker.

(R. Chelbo): Even the opinion that permits bread of a Nochri baker, permits only where there is no Yisrael baker.

(R. Yochanan): Even the opinion that permits bread of a Nochri baker permits only in the field, but not in the city, due to intermarriage.

Eibo was eating bread of a Nochri in the field, outside the border of the city.

Rava: Do not recite teachings in his name, for he is eating bread of a Nochri.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 17:12): Even though Chachamim forbade bread of Nochrim, in some places they are lenient and buy bread of a Nochri baker where there is no bread of Yisrael uv'Sadeh (and in the field), for in She'as ha'Dechak.

Ran (Avodah Zarah 14a DH Rebbi): R. Chelbo says that the lenient opinion permits bread of a Nochri baker only where there is no Yisrael baker. R. Yochanan says that he permits it only in the field, but not in the city. Some say that R. Yochanan refers to R. Chelbo, and is stringent to permit only in the field where is no Yisrael baker. Some say that R. Yochanan's teaching is independent. He always permits in the field and forbids in the city. We can be more lenient and say that he agrees to R. Chelbo's leniency. Since the Isur of bread is mid'Rabanan, and the Isur did not spread through Yisrael, we are lenient. However, the Rambam connotes that the two versions argue like my first explanation. He ruled leniently, and did not mention the field at all.

Beis Yosef (DH v'Chasav Avi): Why does the Ran say that the Rambam is lenient, and does not mention a field? Our text of the Rambam permits only where there is no bread of Yisrael and in the field, for in She'as ha'Dechak. He requires both, like the first explanation; he rules (even) like R. Yochanan to be stringent. The Rashba rules like the last explanation, to be lenient.

Bach (5 DH v'Chasav): Some texts of the Rambam says 'v'Kosher', and do not say 'uv'Sadeh'. Even texts that say 'uv'Sadeh' mean 'or in a field'. This illustrates a case of She'as ha'Dechak. According to the Beis Yosef, the Rambam should have explicitly said that the Heter is only in a field. Also, why did he say 'there are places' if the Heter is only in a field? Rather, he permits bread of a Nochri baker even in the city. Also Isur v'Heter explains this text of the Rambam like I said.

Rashba (35b DH Amar): The Yerushalmi forbids bread of a Ba'al ha'Bayis Nochri even if one has nothing to eat, unless it is Piku'ach Nefesh, e.g. he fasted a lot. Maharam says that he fasted three days, The Ramban says that Ge'onim permitted bread of Nochrim on Shabbos if one has no other bread, for one may not fast on Shabbos. I said that in such a case (that there is no bread of Yisrael), we permit bread of a Nochri baker even on a weekday! The Ge'onim permitted even bread of a Ba'al ha'Bayis if there is no baker. My Rebbi (R. Yonah) says that people who lodge with Nochrim may not eat bread of a Ba'al ha'Bayis. Rather, he must buy from a babker. One who eats bread of a Ba'al ha'Bayis, even not in the Nochri's house, even not in the city, where there is no concern for intermarriage, transgresses Beis Din's decree. Where there is a bread of Yisrael, even bread of a Nochri baker is forbidden. The Heter is only when this is a life essential, like the Yerushalayim says.

Mordechai (Avodah Zarah 830): R. Shamrayah permitted bread of a Nochrim for one who is on the road. He brought a proof from the Mishnah (Demai 3:1) that we may feed Demai to Aniyim and guests, even though there is an Isur mid'Rabanan.

Tosfos (6b DH b'She'as): Rashi says that 'in pressed circumstances' is in famine years. This is wrong. Below (9b), we say that even Rabanan admit in famine years. Rather, 'in pressed circumstances' refers to one who touched a Nidah in me'Es la'Es (before she saw blood) and engaged in a small amount of Taharos, and after the ruling he engaged in much Taharos. This is unlike famine years, since here, initially it was not She'as ha'Dechak, and he did much Taharos through the mistaken ruling. Or it was She'as ha'Dechak because the questioner went away, and it would be a great exertion to pursue him.

Question: In a certain city they normally sell wine slightly cheaper to Nochrim than to Yisre'elim, but one year wine was expensive and there is barely enough Kosher wine to drink that year. Can we permit wine in a barrel that a Nochri touched through Nitzuk (a flow)?

Answer (Terumas ha'Deshen 204): Seemingly, we cannot, even though the Rosh wrote in the name of Maharam, and also the Mordechai said, that regarding a big loss the custom is like R. Tam, that Nitzuk is not a connection. Regarding a small loss the custom is like Rashi, that Nitzuk is a connection. One Gadol wrote that any case of a barrel of any size is called a big loss, i.e. but not what is in a bottle or jug. However, I say that here even a big barrel is called a small loss, for one can sell to Nochrim for a slightly lower price. We say that Medame'a is a small loss, for one can sell it to Kohanim, even though this is for a lower price. We do not distinguish between a large or small quantity.

Suggestion: Since the Tzibur has a limited amount to drink, it is considered a big loss.

Rejection: Chachamim rejected R. Eliezer's proof from a girl who did not see blood for three periods, for it was She'as ha'Dechak. I.e. there was a famine, or she had touched much Taharos. Rashi said that in famine years it is She'as ha'Dechak even if she touched a small amount. Rashi did not want to say that it was a famine year and there was a shortage of Taharos for those who eat Taharos. This shows that we do not permit due to this. Rather, we permit only due to a loss, for the Torah is concerned for the money of Yisrael. However, Rabanan were lenient about Yayin Nesech nowadays in many areas, so perhaps we are not so particular to be stringent. Also, here if we call it a small loss because one can sell it to Nochrim, and therefore we call it Yayin Nesech, this would lead to a leniency. Several Rabanan forbid selling to Nochrim wine forbidden due to contact with a Nochri.

Poskim

Shulchan Aruch (YD 112:2): In some places they are lenient to buy bread of Nochri bakers, where there is no Yisrael baker, because it is She'as ha'Dechak. Some permit even where bread of Yisrael is available.

Bach (5): The Mordechai brings from the Rosh that one may eat bread of a Nochrim where bread of Yisrael is not found. This is the custom in all these lands.

Shach (6): This is unlike cooked vegetables (Siman 113). There is no place where we permit them. This is because bread is the staple of life. This is why we permit when the Yisrael merely throws a chip of wood into the fire (and the Nochri bakes it. We do not call this Bishul Akum, i.e. food that Nochrim cooked.)

Shach (7): They did not forbid due to Bishul Akum because the Tzibur could not bear this decree. This does not apply to bread of Yisrael that a Nochri baked, therefore it is forbidden.

Yabi'a Omer (6 YD 4): A question of Isur v'Heter arose in which the Shulchan Aruch and Acharonim permit only to avoid a big loss. The Rav permitted based on the situation. Later, the questioner became richer or the item cheapened, and it was no longer a big loss. Zivchei Tzedek (35:15) cites a case in which they slaughtered on Erev Shabbos what was needed for Shabbos. There was a question about the animal, and Chachamim permitted it due to She'as ha'Dechak. Later, they slaughtered many animals and meat was plentiful. What we previously permitted remains permitted. It would look ludicrous to forbid it. It seems that the same applies here. The Rema, at the beginning of Toras Chatas, said that whenever we permit due to a big loss, letter of the law it is totally permitted, but some Acharonim were stringent. In She'as ha'Dechak or to avoid a big loss, we leave letter of the law in place. The Rashba says that if a Nochri baker baked bread when there was no Yisrael baker, it is permitted. If a Yisrael bought the bread and then a Yisrael baker came, it is still permitted. Bread of a Nochri baker was forbidden only due to an enactment. It was permitted when there is no Yisrael baker. Chachamim did not make their words ludicrous, that on the same table bread was permitted and forbidden and permitted again.