I'm guessing it means the second half of extra time in a game of soccer/football, partially based on the fact that search results for it seem to be soccer-related.
Or it could just be what your juice replies to you with when you say good day to it.
The fourth half is indeed weird. But in the Netherlands we have an expression called "de derde helft" (the third half). That means when you have played, for example, a game of football (consisting of two halves) you have some beers afterwards. Some players excel in the first and second half, others excel in 'de derde helft'
This sentence isn't nonsensical. Four halves is two (4/2 = 2). It is entirely possible to have four halves of something. An example given above was apples. If you have two apples and cut both of them in half, you have 4 halves. Say the fourth half was odd, and someone wanted to make a comment about it. The fourth half is interesting. Not nonsensical at all.
I collect American coins (for real). You see pennies or quartes nowadays, but not many "halves". More exciting to collect them. There are different types of them (as $1 coin). So, if I see six, and I prefer the forth which has been shown to me, then this sentence make sence.