We are in chapter two, mishnah 8. The mishnah brings three things that Rabbi Akiva said: "Sheloshah devarim amar Rabbi Akiva, al shenayim hodu lo ve'al echad lo hodu lo" - the Sages agreed with him about two of his statements, and about one they did not agree. Let us go through all three in order.
The first item - "al sandal shel sayadim shehu tamei midras":
This footwear, quite literally the sandal of lime workers, is susceptible to midras tumah. To understand this we need some background, which we have already encountered in this tractate: one of the categories of tumah is called 'midras'. A vessel designed to bear a person's weight - a saddle, a chair, a rug or shoes - receives tumah from four people whose tumah emanates from their own bodies: a zav, a zavah, a niddah and a woman after childbirth. These transmit tumah by way of midras, meaning even without direct contact: if one of them stood with his shoes on the vessel, and even if he was wearing socks, the shoes become tamei. And not only do they become tamei, they become a principal source of tumah of the highest order. This is an exceptional law.
The question is whether this law applies in our case. The rule is that midras tumah applies to anything designed to bear a person's weight, including shoes. But the footwear before us is not really shoes at all, but a sort of wooden clog meant to protect lime workers. Lime is a highly caustic material that burns the skin, and for this reason the workers needed protection for their clothing and their skin. True, they do walk around in this footwear, but it is not designed for ordinary walking.
Rabbi Akiva's view: the workers do in fact walk back and forth in it, and once they put the equipment on they keep it on even indoors and walk around in it. Since that is so, its purpose is not only protecting the skin but walking as well, so it turns out that it is designed to bear a person's weight and is therefore subject to midras tumah.
The Sages' view: its primary purpose is protecting the skin, and according to some of the early commentators - protecting the clothing or the shoes underneath it. These wooden boards are not designed for walking at all, but are purely protective equipment, and not meant to bear a person's weight. And even if one walks in them a bit, that is only incidental and not their purpose.
Nevertheless, the halachah is that the sandal is tamei with midras tumah. That is the first of the three items.
The second item - "ve'al shirei tanur arba'ah":
If an oven broke and what remains of it is four handbreadths high, it is susceptible to tumah. The mishnah adds: "shehayu omrim sheloshah" - initially the Sages said that the minimum measure for susceptibility to tumah is three handbreadths, "vehodu lo" - and they retracted in the face of Rabbi Akiva's position and said that the measure is four.
Here too the background is involved. Vessels are susceptible to tumah, meaning they are fit to receive it. Ovens in the times of the mishnah were fairly simple: large earthenware cylinders attached to the ground with a hole at the bottom. The fuel and the fire were placed at the bottom, the heat rose upward, and sometimes there was a shelf on the side. The standard size for making a new, small, ordinary oven is four handbreadths high. True, if the oven broke a little at the top and three handbreadths remain, it will still work, but the starting point for all ovens to begin with is four handbreadths.
A vessel that breaks ceases to be susceptible to tumah, and the question is how much of it must break in order for it to lose its status of susceptibility. Consider a very large oven that was ten or nine handbreadths high, and now most of it has broken - is it still considered that same oven, susceptible to tumah?
The Sages' initial view: as long as three handbreadths remain, since an oven functions at that height, it still fills its entire role and is therefore susceptible to tumah.
Rabbi Akiva's view: when people make ovens, they do not make them less than four handbreadths to begin with. Therefore an oven that broke, even if most of it broke, is tamei - provided that four handbreadths remain, and if not, it is tahor.
And the halachah was decided like him.
The third item - "ve'echad lo hodu lo":
This is the opinion of Rabbi Akiva that was not accepted: "al kisei shenitlu shnayim michipuyav zeh betzad zeh". According to the Bartenura and most commentators, even though the word "chipuyav" is the same word we came across at the end of the first chapter regarding a bride's chair, here we are dealing with an ordinary chair that has lost two of its boards. The basic picture is something like a wooden box that serves as the base of a chair, and sometimes a backrest is added to it afterwards for comfort. In practice it resembles a stool more than a chair.
The wooden boards running along its top are the seat. The usual arrangement was to lay three boards across the width, each one about three or four inches wide. As long as all three boards are in place, the chair is whole and comfortable. Even if one of them breaks, you can still manage on two, and even if the two outer boards break and only the middle one remains, you can manage on one. But if two boards adjacent to each other are removed, such as the outer one and the middle one, only a narrow board is left. Now it is no longer comfortable to sit on, and it ceases to function as a chair, and therefore we must say that it has lost its status as a vessel that can receive tumah: it was a vessel in its capacity as a chair, and now it is a chair that nobody sits on, genuinely broken, and so its ability to receive tumah has lapsed.
To this Rabbi Akiva says: even if the wooden box can no longer be used as a chair, it can be used for storing things. A person does not throw it out; he keeps it as a kind of small box with a broken lid, in which things can be placed. Therefore it still receives tumah, because it has a receptacle. The view of the Sages, and this is the halachah, is that it no longer serves its primary function. When this item was manufactured, it was made to be a chair to sit on, not a box for storing things, and therefore once it is broken as a chair it no longer receives tumah.
In the language of the Mishnah: "ve'echad lo hodu lo" - there was one item on which they did not agree with Rabbi Akiva, "al kisei shenitlu shnayim michipuyav" - two of the three boards of the seat broke (some explained "chipuyav" from the word for a spoon's bowl, like the ridges on a key), "zeh betzad zeh" - two that are next to each other, so that a large gaping hole is left in the seating area. "sheRabbi Akiva metamei" - it still receives tumah, since even though one cannot sit on it comfortably, things can be stored inside it.
There is another way to explain this: the matter does not depend on storage at all, but rather that even though one cannot sit on it comfortably, one can sit on it with difficulty when necessary. Just as we saw in the previous chapter, that people sat on the ledges of kneading troughs, so too one can sit here on the thin four-inch strip that remains of the broken chair. According to this explanation, Rabbi Akiva is saying that the chair still functions.
"vaChachamim metaharin" - the Sages say that it does not receive tumah. According to the first explanation: it was not made for storage, and the fact that things can be stored in it is not enough to change that. And according to the second explanation: the fact that one can sit on something is not enough to make it a chair, since it is broken, and therefore it does not receive tumah. That is the essence of the Mishnah here.