He is never unavailable
He is always available
If you want to look at this from the perspective of whether a doctor is available at a certain time, both of these sentences mean the same thing. And the first one feels rather easy to understand.
However, when you use the word "not" twice in a sentence, many English speakers will find the sentence to be challenging. Such questions are easy to say incorrectly. You can use it correctly, and there is nothing technically wrong (or "rule-violating") with the grammar.
He is not unhappy
He is happy
These phrases may be slightly different. Although the prefix "un" usually means "not", this is a bit of a special case. The word "unhappy" usually means "sad". So, "not unhappy" mean simply mean that a person is "not sad", but could mean that the person is ambivalent/neutral, and is neither happy nor sad.
There are cases where it is appropriate to use a phrase like "not unhappy". Because this feels a little bit less straight-forward, this sort of phrase may require a little bit more effort for people to process in their brains. Speakers may assist by slowing down their speech, and perhaps speaking a bit louder, to help emphasize the individual words "not unhappy", and pause for a second or two so that the receiver can understand the full implications.
Here is another example:
He is not unavailable
He is available
(When speaking, the syllable "un" might be expressed with extra strength/emphasis.)
If this is advice given to a woman who wants a specific man to be a boyfriend, these might be different. The second sentence suggests that the man has no girlfriend. He is available for such a relationship. The first sentence might describe a man who isn't married (because he would be unavailable if he was married), but who is in a relationship that he's not happy with. An ambitious woman might try to convince him to end the unhappy relationship. So, he's not easily available. In fact, making him available might require a bit of effort. But, he's a lot closer to being available than someone who is happily married.
Note that this example is almost the same as the first example, when I said the sentences did mean the same thing. In this case, the sentences do have a small difference. When people talk like this and there is a difference, the difference is often small. In some cases, there isn't even any difference. In many cases, a person will need to evaluate a situation's possibilities to think about what is being said.
Sometimes different people will interpret things differently.
Example case: I am really into computers, and as an experienced computer programmer who has worked with bit-wise logic, I can easily understand the flipping of meaning caused by logical negation. I find myself to be more prone to correctly use double-negatives than many other people. On rare occasion, I even find a triple negative to be the most precise way to say something. Such a feat usually requires a bit more thought, as I make sure I speak with accuracy and technical precision.
However, my father is a fairly intelligent man who has demonstrated a particular weakness at being able to understand double negatives. When I use them, he says that he doesn't follow, or asks for a re-phrase.
Remember that communication is pointless if the receiver cannot understand what is transmitted. (This is true for both natural speech and electronic communication.) A lot of the other answers have focused on the grammatical rules. I'm pitching in, by adding this important perspective: a lot of times that double-negatives are used, it will cause significant strain for many audiences. This may be entirely appropriate for some scenarios, such as technical specifications which must be interpreted with precision (and is likely to be processed by intellectuals who may work with specific technical concepts, like logical negation, and thinking about which set elements are members of specific sets). For many other scenarios, such technical speech is often more cumbersome than useful. Talking this way is not what many people prefer for casual speech that is easily understood.
Most people can handle some simple double negation, with a bit of challenge. In some cases it is appropriate. In many cases, you may be better off seeing if you can re-phrase the sentence while containing the exact same meaning. In some cases, you can, and doing so will be easier.