Gemara

Beraisa: "V'Rochatz ba'Mayim Besaro" - in a Mikveh;

"Kol Besaro" - the Mikveh must contain the entire body (of an average size person).

Question: How big is this?

Answer: It is an Amah by an Amah by three Amos tall (or any shape of this volume). Chachamim calculated that this is 40 Sa'im.

Pesachim 109a - Rav Chisda: The Revi'is mid'Oraisa (to Mechayev a Nazir for drinking wine, for Tum'as Ohel of blood, and the oil for certain Menachos) is the volume of a box two thumbs by two thumbs by two and seven tenths thumbs:

Eruvin 3b - Rava: All Amos (that Chachamim taught, that are not written in Tanach) are six Tefachim. Some are Sochakos (the hand-width is measured with space between the fingers), some are without spaces.

Pesachim 17b: Liquids in the ground are Tehorim if there is a Revi'is, for (mid'Oraisa) it is a valid Mikveh for Tevilah of small Kelim.

Nazir 38a: One may be Tovel Kelim in a Mikvah containing a Revi'is of water.

Chachamim abolished this law (and require 40 Sa'im).

Avodah Zarah 75b: Regarding the Kelim taken from Midyan it says "V'Taher" and "B'Mei Nidah Yischata";

Had it said only "V'Taher", one might have thought that Tevilah may be in any amount of water. "B'Mei Nidah" obligates water that a Nidah could immerse in, i.e. 40 Sa'im.

Mikva'os 5:1: A Mayan (spring) of any size is Metaher.

Rishonim

Rif (Pesachim 23b): A Mikveh must contain the entire body, i.e. an Amah by an Amah by three Amos; this is 40 Sa'im.

Note: The Rif shows how it follows that a Revi'is is 2 by 2 by 2.7 thumbs; we give a simplified calculation. One Se'ah is six Kavim, which is 24 Lugim, which is 96 Reviyos, so 40 Sa'im is 3840 Reviyos. This equals three cubic Amos, so each cubic Amah is 1280 Reviyos. An Amah is six Tefachim, which is 24 thumbs, so a cubic Amah is 24x24x24 thumbs. If we divide this by 1280 (which is 8x8x20), we find that one Revi'is is 3x3x1.2 = 10.8 cubic thumbs, which equals 2x2x2.7.

Rambam (Hilchos Mikva'os 4:1,2): One may be Tovel in "Mikve Mayim", any gathered water, as long as it contains the entire body at once. Chachamim estimated that this is an Amah by an Amah by three Amos, which is 40 Sa'im.

Rambam (ibid. 9:8): A Mayan of any size is Metaher.

Rosh (Hilchos Mikva'os Sof Siman 1): A Mayan of any size suffices for Kelim, but 40 Sa'im are needed for Tevilah of a person.

Poskim

Shulchan Aruch (YD 201:1): A Nidah does not become Tehorah until she is Tovel her entire body at once in a Mikveh or Mayan with 40 Sa'im.

Shach (2): It suffices if there are 40 Sa'im in the entire Mayan.

Gra (DH 1): Most explain that a Mayan of any size is Kosher for Tevilah even for people, unlike the Rosh. Nevertheless one should be stringent regarding a mid'Oraisa law.

Aruch ha'Shulchan (10): If one was Tovel in a Mayan and is unsure whether or not it had 40 Sa'im she may be lenient, for it is a Sefek-Sefeka (even if it was less, perhaps the Halachah follows the opinion that any amount is Metaher), and it overrides Chezkas Tum'ah.

Shulchan Aruch (ibid.) The 40 Sa'im are an Amah by an Amah by three Amos. The Amah is six Tefachim and half a finger. In all this is 44,118.5 (cubic) fingers.

Source (Beis Yosef DH v'Chosav): Rava (Eruvin 3b) requires Amah Sochek whenever this is a stringency.

Divrei Chamudos (Nidah 10:45): The Beis Yosef wrote that there are 41,732 (cubic) fingers in a Mikveh. This ignores the extra half finger (in each Amah). In Pesachim, also the Rif, Rashi, Tosfos and Rashbam ignored it. This is because there the discussion is a Revi'is, which is mid'Rabanan.

Rebuttal (Shach 6): The Gemara there discusses Revi'is of the Torah! The Gemara always says that the Shi'ur of a Mikveh is 40 Sa'im. According to Divrei Chamudos, when we add the extra half fingers there will be more than 40 Sa'im! Rather, regarding Mikveh we normally discuss Amos and fingers that are Sochakos. The Shulchan Aruch gives the Shi'ur in fingers that are not Sochakos.

Aruch ha'Shulchan (2): The Gemara did not specify whether or not the Amos for Mikveh are Sochakos. Since Nidah entails Kares we are stringent.

Shulchan Aruch (ibid.): The Mikveh can be wider and shallower, as long as she can be totally covered by the water. The Mikveh must be big enough so it will hold 40 Sa'im even when the person enters (raising the water level).

Beis Yosef (DH Amrinan): Anything less than 40 Sa'im does not suffice.

Shulchan Aruch (120:1): If one bought Kelim for food from a Nochri, he must be Tovel them in a Mikveh or Mayan with 40 Sa'im.

Beis Yosef (DH v'Chosav b'Sefer and DH u'Mah she'Chosav Rabeinu): SMaK, Tosfos and the Rashba say that even though a Mayan of any size is Metaher, 40 Sa'im are required to be Mekadesh Klei Nochrim for use by Yisrael. They learn from "b'Mei Nidah" (Avodah Zarah 75b). The Tur disagrees, for we learn about this Tevilah from a Teme'ah (Nidah), so it is unreasonable to be more stringent about it than about Nidah!

Question (Beis Yosef, ibid.): The Tur should be concerned for the Rosh's opinion that also a Nidah requires 40 Sa'im even in a Mayan!

Answer #1 (Darchei Moshe): The Tur questions SMaK, who is more stringent about this Tevilah than about Nidah.

Objection (Taz 2): Surely, also SMaK requires 40 Sa'im for a Nidah even in a Mayan!

Answer #2 (Taz 2): The Tur means that it is unreasonable to be more stringent about this Tevilah than that of Tum'ah, i.e. of Kelim.