The phrase “hare's breath” is a malapropism which is commonly used by people who mean “a hair's breadth.” While “hare's breath” at least makes sense, other variations of this malapropism, such as “hair's breath,” are totally illogical. The confusion about this phrase reflects a common problem with homophones in the English language: because “hair” and “hare” and “breath” and “breadth” sound so similar, people sometimes mix the words up when they are writing, especially if they have never seen the phrase written out before.
Properly, the term “hair's breadth” is usually used to reference a narrow escape or a closely missed opportunity. The breadth of a hair is indeed quite small, and a hair's breadth between two objects or event would be a very narrow margin. One may also hear phrases like “came within a whisker,” in a variation on a hair's breadth. An example of the correct usage of this phrase is: “we came within a hair's breadth of winning that contract, but the other company underbid us.”
Some people attempt to defend “a hare's breath” by suggesting that it refers to something which happens very quickly, just like the breath of a rabbit. However, since all of the colloquial uses of this idiom revolve around something which almost happens, rather than something which happens very quickly, this explanation does not hold very much water, and it is probably a backformation intended to justify a common malapropism. Unless one is talking about the breathing habits of hares, “a hare's breath” is incorrect, and a “hare's breadth” would also be incorrect, unless one is using a hare as a unit of measurement, which seems unlikely.
In addition to “hair's breath,” another creative variation on this commonly misspelled idiom is “hair's breathe.” Both of these usages are nonsensical in addition to wrong, unless hair has some hitherto unknown biological properties.
For English language learners, homophones can be especially frustrating, as the existence of multiple words which sound the same but have different meanings and spellings can be very confusing. All of the variations on this idiom sound the same when they are spoken aloud, but they have very different meanings. Slipping up can be viewed as an elementary mistake by an experienced user of English, and it tends to detract from the overall quality of a written communication. Because homophones can be tricky, it is a good idea to ask someone to look over a piece of written material before publishing it or sending it to someone else, to make sure that glaring errors such as “hare's breath” or “hair's breath” do not remain.