Something that is palatable is acceptable to one’s sense of taste—literally or figuratively. If it's palatable, then you can put up with it — whether it's leftovers or a mediocre made-for-TV movie.
The palate is the roof of the mouth, the combination of structures that separates the mouth from the nose. Early anatomists believed that the sense of taste was located in the palate, and, just as taste is metaphorically expanded to include sensibilities beyond the experience of food and drink, so palatable can be used to describe phenomena beyond the culinary. And, while palatable can mean pleasing or agreeable, it generally means merely tolerable—edible, rather than delicious.