Why doesn’t water flow out of a small hole from the bottom of a bottle?

Water will leak out of the hole until the pressure just inside it equalizes with that just outside (it will actually stop a bit before that due to surface tension, which can hold back a small pressure difference when the water surface bulges out through the hole). If there is, say, a 10 cm column of water above the hole, that contributes an excess pressure over atmospheric of 1 kPa, which will push water out until the air space above the water expands by about 1%, at which point the decrease in air pressure cancels the hydrostatic pressure from the water.

It the hole were much bigger, air would be able to get in there, and, glug glug, the water would all come out. But because of surface tension, you can’t simultaneously get water to go one way and air the other through a small hole.