There are some people with mental 'problems' who dislike being around people and get anxious. For these people it is possible to be happy when they're completely alone.

But, for a healthy human being, being alone will not only make them unhappy but it will drive them crazy. This is mostly because of the way evolution works and how the human psychology corresponds with it.

The most important thing for evolution is other people of course, without them you can never pro create.

So the final answer: a completely healthy human being CAN NOT be happy when he/she is alone for a long long time. But, if a person has a mental disorder he/she can be