Eduyos, Chapter 8, Mishnah 5. This mishnah records two separate testimonies: one concerning marital law, and one concerning matters of purity in the Beis HaMikdash.
The first testimony - permitting a woman to remarry on the word of a single witness:
"Heid Rabbi Akiva mishum Nechemiah ish Beis Deli, shemasi'in ha'ishah al pi eid echad" - Rabbi Akiva testified in the name of Nechemiah, a man from the town of Beis Deli, that a woman is permitted to remarry on the strength of a single witness who testifies that her first husband died.
This matter was already taught at the beginning of the sixth chapter, and we must ask how this testimony differs from the testimony of Rabbi Yehudah ben Bava there. For simplicity's sake we will assume that there is no additional novelty here, only a repetition, though there are other possible answers that draw out a nuance between the two. The bottom line, as there, is what the Sages said: by strict law, since the woman is presumed to be married, two valid witnesses would be required in order to act in a matter of forbidden relations and permit her to remarry. The Sages, however, were lenient here in order to prevent her from becoming an agunah, left stuck and unable ever to marry, and therefore they accepted the testimony of a single witness, even a witness who would not ordinarily be acceptable.
The reason for this is that two assumptions counterbalance the force of the presumption:
The truth is destined to come out: If the testimony is false, the husband is liable to show up one day and ask, "Where is my wife?" - for if he is alive, the matter will eventually come to light. And we have an established principle that a person does not lie about something whose truth is destined to be revealed.
The severity of the consequences: The Sages imposed a series of severe consequences should she be mistaken - she loses her kesubah, any children she bears will be mamzerim, and so forth. On that basis they assumed that a woman would be extremely careful and thorough in verifying that her husband really did die, and that she herself is certain of it, before remarrying.
The second testimony - bones found in the wood chamber:
"Heid Rabbi Yehoshua al atzamos shenimtze'u b'dir ha'eitzim" - his testimony concerned human bones that were found in the dir ha'eitzim, the wood storehouse. This refers to the Chamber of the Wood Storehouse, one of the four chambers in the Ezras Nashim, at the entrance to the first section of the Beis HaMikdash complex. On the northeastern side, the first chamber on the right as one enters, it was designated for storing the wood used for the arrangement of fire on the Altar. Once human remains were found there, the question arose: must we be concerned that additional remains lie there deeper down, and should we begin searching for them?
On this Rabbi Yehoshua testified as to what the Sages ruled: "Melakeit etzem etzem v'hakol tahor" - and this contains two rulings:
"Melakeit etzem etzem": One removes the individual bones that were found, one by one, and clears them all away, and that is sufficient. There is no need to dig deeper and check what else might be found there.
"V'hakol tahor": We do not assume that anything became contaminated inadvertently through contact with these bones - not people, not the various vessels of the Beis HaMikdash, and not offerings.
The reason for the second ruling is straightforward: this is a case of doubtful tumah in a public domain. The wood chamber is considered a public domain, since people are commonly found there and come and go, three or more - and a doubtful case of tumah in a public domain is ruled pure. Therefore everything retains its purity.
As for the first ruling, it is explained in the Gemara elsewhere that they did in fact consider searching and checking what other bones might be found in other places. Rabbi Yehoshua responded: there was the Flood, there was Nevuchadnetzar, and bones are liable to be found anywhere. Surely those who built the Temple complex checked this in their day, and from here on we may assume that there are no bones there and rely on that assumption. We do not disgrace Jerusalem and declare the entire city one large cemetery, as had been suggested. Therefore it is enough to remove those few individual bones, and everything else is completely pure.
Another matter is worth noting, although its connection to our discussion is not clear: the Gemara, both in the Bavli and in the Yerushalmi, relates that it once happened that a man discovered a floor tile out of place in the Chamber of Wood, and there was an indication in this that it was the spot where the Ark of the Testimony was hidden away. Before he managed to reveal the matter, he died; and according to the version in the Yerushalmi, he began to dig and a fire came out and consumed him. What emerges from here is a separate, independent reason for not wanting to dig in the Chamber of Wood: because hidden treasures lie buried beneath it.
In summary: the halachah was decided in accordance with both of these testimonies - a woman is permitted to remarry on the strength of the testimony of a single witness that her husband has died; and when bones are found in the area of the Temple, we assume that there are no others besides them, we gather them up and remove them, and everything else is tahor.