The world Inception takes place in is very similar to current-day Earth. It paints a very familiar picture and is meant to either be present-day or the very near future, as the technology level shown, outside of the dreamshare technology, is at today's level.
The one major difference in this world is that at some point, the military created the technology to allow people to share dreams and enter each other's subconscious. This was developed as a training program, though the criminals in the film show us that there is an underground of people who have started to use the technology to steal secrets. In the film, they are given the task of actually influencing someone's decisions, though none of them know whether or not it's ever been done before.
Arthur works in this criminal underground. It is unknown what his background is, though his comfort with the technology and knowledge that it came from the military can be used to infer that he was involved in the military application of the technology at some point.
When using the dreamsharing technology, many of the characters are shown to have totems--an item that only they know the weight or movement of--so that they can tell whether or not they're in someone else's dream. This is necessary because in a dream, characters don't remember how they arrived in that new place. Keeping track of reality is a very real problem for them. Arthur's totem is a red loaded die.
The importance of keeping track of reality is there because there are two ways to leave the dream; either the clock runs out on the machine or you die. Losing track of reality can lead to attempting to leave a dream when you are in fact in reality, which happened to someone they knew.
Arthur has been out of the Matrix for four months. Having come to rely on his totem to tell dreams from reality, he began to struggle without it, often reaching for it and slowly beginning to question if the reality he was brought out into wasn't a dream. While in rehab, this behavior was noticed and he's been kept under psychiatric watch for his own safety after he finally admitted to his struggle.
He's spent the last month pretending that he finally believes this to be reality so that he would be released. Though he's still somewhat unsure if it is or not, he does think that it could be. He's making an effort to be cautiously optimistic, while still being incredibly paranoid and generally uncomfortable about the entire situation.
▲ ABBREVIATED DOSSIER
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