Gemara

(R. Yehoshua ben Levi): "Ani ha'Ishah ha'Nitzeves Imchah ba'Zeh" -- [Eli was standing because] one may not sit within four Amos of someone praying.

Rishonim

Rambam (Hilchos Tefilah 5:6): One may not sit to the side of someone standing in Tefilah, until he distances four Amos.

Hagahos Maimoniyos (30): R. Simchah found in the name of Rav Hai Gaon and R. Natrunai Ga'on that this is if he is idle. If he is engaged in Tefilah or other matters of song and praise, he need not distance. Our Rebbeyim ruled like this.

Rosh (5:8): One may not sit within four Amos of someone praying. One must distance four Amos, whether in front, in back and to the side. The Gematriya of "ba'ZEH" (i.e. without the Beis) is 12, which alludes to four Amos in each of these three directions. Teshuvas ha'Ge'onim says that this is only if he sits idle. If he is engaged in Tefilah, e.g. Keri'as Shema and its Berachos, he may sit.

Rashi (31b DH Imcha): "Ha'Nitzeves Imchah" connotes that also he was standing with her.

Tosfos (31b DH Imcha): "Imchah" (with you) connotes within four Amos, for it is written with a "Hei". (Its Gematriya is five.) This hints that he would not sit within four Amos, rather, only in the fifth Amah from her.

Poskim

Shulchan Aruch (OC 102:1): One may not sit within four Amos of someone praying.

Taz (1): The Tur learns from "Imchah" that Eli was standing within four Amos of Chanah. Since he thought that her lips are moving because she is drunk, but not because she is praying, why was he standing? Rashi (Shmuel 1:1:13) says that people did not used to pray quietly. We can say that he thought that she is praying quietly because she is drunk. Also, even if he did not know that she is praying, she stood to pray within four Amos of him only because he was standing. According to Tosfos, we can say that he was sitting more than four Amos away.

Mishnah Berurah (1): One may not lean even a little, so that if the support would be removed he would not fall. One may be lenient in pressed circumstances.

Mishnah Berurah (2): Even when he is on a diagonal, four Amos suffice. We do not require four Amos and their diagonal (i.e. of a square whose side is four Amos). The Chayei Adam and Hilchos Ketanos say that if one prays on one side of something 10 Tefachim tall and four Tefachim wide, one may sit on the other side, for it divides the Reshus. Perhaps one may not pass on the other side if it is not so tall that the one praying cannot see on the other side. Also the Eshel Avraham (90:38) forbids passing in an Almemra (an enclosure around the Bimah), even though it is a separate Reshus, but one may sit in a separate Reshus, even within four Amos.

Mishnah Berurah (3): Even if the one praying is in supplications after Shemoneh Esre, as long as he did not step back, one may not sit in front of him.

Shulchan Aruch (ibid.): One must distance four Amos, whether in front of him or to the side (Rema - or in back).

Taz (2): The Mechaber and Tur did not mention in back. They hold that the Siman of 12 refers to four Amos in front, and four Amos on each side, but in back is permitted. We learn from "v'Eli Shomer Es Piha." He could see her mouth only if he was in front of her or to the side, but not in back of her. The Rosh forbids in back. He must hold that both sides count like one. It is astounding that the Tur did not bring his father's opinion, since he argues with him. Also Tosfos and the Mordechai forbid in back, and so rules the Rema.

Note: The Maharshal deleted "or in back" from the text of Tosfos 31a DH mi'Kan.

Magen Avraham (1): According to the Rosh, the Siman should be 16, to show that four Amos in every direction are forbidden! The Bach answers that Chanah prayed by the wall, so there was no need to distance in front of her, but if one prays in the middle (of the room), one may not sit in front. I say that one may not sit in a circle of four Amos from the one praying. It is known that (the circumference of) such a circle is 12 Amos, like it says in Sukah 8a. Even though here four Amos were forbidden in every direction, so the diameter was eight (and the circumference was 24), the Siman teaches about Kav ha'Shaveh (i.e. a semi-circle, e.g. in front of or on one side of the one who prays. Indeed, there is another semi-circle, in back or to the other side, in which one may not sit.) According to the Yesh Omrim (at the end of this Sa'if, who forbid in front as far as one can see), there is no question, for we do not discuss in front.

Note: I do not understand this, for the Yesh Omrim are stringent only directly in front. They agree that in any other direction, within four Amos are forbidden. If so, this is almost a full circle. Its circumference is almost 24. It is far more than 12!

Mishnah Berurah (4): The Acharonim forbid also in back.

Kaf ha'Chayim (4): Some say that the text of the Tur should forbid also in back. It is a printing mistake. The Acharonim forbid in back.

Kaf ha'Chayim (5): One may sit in a high place that is a separate Reshus, even if it is within four Amos. If there is a glass wall in front of the one praying, one in front of him must not face the one praying, lest he distract him.

Shulchan Aruch (ibid.): If he is engaged in matters of Tefilah, even in (Korbanos, e.g. the Mishnayos of) Perek Eizehu Mekoman, he need not distance. Some permit one who is engaged in Torah, even though it is not connected to Tefilah.

Beis Yosef (DH u'Mah she'Chasav Rabeinu): The Tur says that the Ge'onim forbid one engaged in Torah, for they mentioned only one engaged in Keri'as Shema, its Berachos and similar matters. The Tur does not exclude someone engaged in other matters of Tefilah. The Ge'onim said 'and similar matters'! Hagahos Maimoniyos and Terumas ha'Deshen (3) say like this. Perhaps even one who is saying Eizehu Mekoman may sit. Since the custom is to say it before Tefilah, it is like a matter of Tefilah. R. Yonah (23a DH Ani) permits one reading 'Keri'as Shema or verses or any matter.' R. Yerucham forbids one who says Korbanos or Pesukei d'Zimrah, but brings an opinion that permits Pesukei d'Zimrah.

Beis Yosef (DH u'Mah she'Ta'an): The Tur says that the Isur to sit is because it looks like the one praying is accepting Ol Malchus Shomayim, but the one sitting is not. This applies even if the latter is learning Torah. One can reject this. If one has no occupation other than Torah, he need not interrupt Torah to pray. If so, Torah is greater than Tefilah!

Taz (3): If one says Divrei Torah not connected to Tefilah, the Tur obligates him to stand, for it looks like he is not accepting Ol Malchus Shomayim, unless he is engaged in Keri'as Shema and its Berachos. This is difficult. If so, it should be forbidden to sit silent within four Amos of someone saying Keri'as Shema, which is the primary acceptance of Ol Malchus Shomayim! Tefilah is merely a request for mercy. Also, how does distancing four Amos remove the concern? What is the source for the Tur's reason? We learn from Chanah! Perhaps it is because the place where one prays is Kodesh. Therefore, one within four Amos must show honor. If one sits without engaging in Kedushah, he treats the area like Chulin. Therefore, one must stand or engage in Tefilah or Torah. This is explains the opinion that permits one who is engaged in Torah. The Beis Yosef says that this is only if he says the words, but not if he merely thinks Torah, which has less Kedushah. This is correct.

Kaf ha'Chayim (1): The Zohar forbids because the place is Kodesh, like the Taz. This is why the Gemara mentions within four Amos of Tefilah, and not within four Amos of someone praying.

Beis Yosef (DH u'Mah she'Ta'an): The Tur says that presumably, Eli was learning (and even so, he needed to stand), for a Chacham may not stand near filth (since he is always learning). This is not a proof. It is not forbidden to be idle from Torah, just we assume that a Chacham cannot refrain. Even if you will say that he may not be idle, perhaps it suffices to think Torah, and this does not permit sitting.

Mishnah Berurah (5): Similarly, one saying Pirkei Avos at Minchah on Shabbos may sit near someone praying.

Mishnah Berurah (6): It is good to be stringent if he is learning Torah, if it is not pressed circumstances. One need not be stringent if he is in back.

Kaf ha'Chayim (7): Whenever the Mechaber is stringent, and says that some are lenient, he holds that one may be lenient in pressed circumstances, e.g. he has no place to sit outside of four Amos, or he cannot stand due to weakness or he is concerned for haughtiness, or lest he pressure the one praying to rush his Tefilah. He should say some Tehilim. If he is idle, this is forbidden.

Mishnah Berurah (7): If he is merely thinking, even if he is engaged in Tefilah it is forbidden, lest people think that he is not accepting Ol Malchus Shomayim.

Kaf ha'Chayim (9): Merely thinking Divrei Torah does not help, but if he hearkens to Chazaras ha'Shatz he may sit, for he is obligated to hear it, and one who hears is like one who reads.

Shulchan Aruch (ibid.): Some say that this is only on the side. Directly in front is forbidden as far as he can see, even if he is engaged in Keri'as Shema.

Mishnah Berurah (8): In front is forbidden for it looks like he bows to him. If so, it is forbidden only to sit, but he may stand, even within four Amos. Perhaps one may not even stand, for this disturbs his Kavanah.

Mishnah Berurah (9): The Pri Chodosh rules like the first opinion. The other Acharonim say that it is good to be stringent. I say that surely one need not be stringent to stand outside of four Amos, and perhaps even within four Amos.

Kaf ha'Chayim (2): One may not stand in front (within four Amos), for it disturbs his Kavanah.