So I'm watching Red Garden (which I'll write about in more detail later) and I'm impressed by the way the characters are written. They actually seem to behave like real people, which is pretty surprising for an anime. This isn't because 'all anime writing is bad', but a fair amount of it is, and even when it's not bad there may be an attempt to write exaggerated or overly dramatic characters. So, in so specific order, here's my list of 'anime with characters that act almost like normal human beings':
Colourful
Anne of Green Gables
Hataraki Man
Wandering Son
Aoi Hana
Honey and Clover
Koi Kaze
Note, there is a difference between 'believable' and 'human beings'. Lots of characters are 'believable' for the shows that they are in, but that's something different. To be fair, most shows aren't trying to achieve this at all.
If I was being really strict I might have to shorten that list further.