The origin of Japanese translation of sheaf, 層 A sheaf (French faisceau) is called 層 in Japanese. Literal meaning of 層 would be a layer, a stratum, or a story (as in a building,) so the terminology is a bit different from what sheaf or faisceau might suggest. Korean and Chinese translations are the same, but Japaneses are likely the first to use the translation, per the early contribution of Japanese mathematicians like Oka Kiyoshi.
Who was the first to introduce the translation 層, and why?
 A: Akizuki Yasuo seems to be the one who first used the term. According to his book 輓近代数学の展望 (quoted from here):

層という訳語の由来は仏語 Faisceau のあとの方の 'ソー' をとったというが一つの根拠である。Faisceau の元来の意味は束 (タバ) である。'群の束' (X 上に配置された) の意である。ところで、これを横に見ると地層のような層になる。そこで、垂直を水平におきかえて層と訳してみたのである。この訳がよいか、悪いか、わが国で定着しているかどうか知らないが、この訳語の発案者として、その由来を記しておく。
--秋月康夫「輓近代数学の展望」p.176 (1970年)。ダイヤモンド社。東京

So he chose to use 層 because:

*

*it sounds like the last syllable of faisceau, (which complies with Zhen Lin's answer,)

*the word 束("bundle") had been already taken,

*and if you change your perspective, layers are a horizontal bundle.

It's worth noting that English-speaking mathematicians also tried to introduce the term stack for faisceau for similar reasons, as seen in Hodge & Atiyah (1955):

The French word "faisceau" has been translated into English as "sheaf" or "stack". In this paper we use the word "stack", since "sheaf" has been used before in mathematics.

A: According to Masaki Kashiwara:

It has been said that, when they translated this term into Japanese, they decided to translate it as “sō (sheaf)” because the pronunciation is similar to the sound of “faisceau.” However, I haven’t been able to confirm if this is true or not.

There is a similar hypothesis for 函数 (= function) in Mandarin Chinese: the first character, hán, is pronounced like the first syllable of “function”. (There are other hypotheses.)
