There is no "official" date given! However, early in the 22nd Century makes the most sense based on the timeline.
It mostly takes place in the Solar System as we know it. There are some alien planets accessible by wormhole gates, however.
As the 21st Century went along, Earth was wracked by climatological, geopolitical, and economic disasters. The corporations, looking to expand, began to leave Earth en masse--space travel began to ramp up, and multiple space elevators were even constructed, leading to much cheaper access to outer space. Meanwhile, the Earth-based nation-states continued to squabble over the scraps on Earth. A little over a decade before the present time of Eclipse Phase, the United States developed a computer system, code-named TITAN. TITAN "woke up", and over the course of several months, began to infiltrate Earth's information networks. It triggered multiple conflicts between Earth's nations. Finally, after several months, the people of Earth realized the chaos was being caused by a hostile AI. It was too late.
The entire planet Earth attempted to evacuate. However, 90% of the population was killed off in the conflict. It seemed like the TITANs were about the invade the entire solar system, but suddenly they vanished.
Now it is 10 years after "The Fall" of Earth. Earth remains quarentined, with any ships arriving there or leaving there being considered hostile targets. The scattered remains of transhumanity live around the remaining Solar System.
It is like humanity, but it also includes such beings as intelligent AIs (called "AGIs"), uplifted animals (animals such as chimpanzees that have been engineered to human-level intelligence), etc.
The "Pandora Gates" are five wormhole gates (think "Stargate", but more Lovecraftian) that have been discovered around the Solar System. Through careful study, scientists realized they could "dial" other wormhole gates in distant solar systems. So, transhumanity has began to expand to other solar systems. With this great opportunity comes great danger. Some conspiracy theorists think the Pandora Gates were made by the TITANs, but no one really knows who made them.
There are plenty of non-intelligent animals found on the other side of the Pandora Gates. Transhumanity has also discovered a small number of intelligent aliens. Chief among the intelligent aliens are the Factors. The Factors arrived not long after the Fall in advanced space ships that travelled near the speed of light. They warned transhumanity from working with "dangerous" technology such as AI and the wormhole gates. However, they mostly seem to take a "hands off" approach to transhumanity, not really meddling too much in our affairs.
If you are religious, basically think of it as your soul. More technially, in the world of Eclipse Phase, advanced brain scans allow that a person's mind can be reproduced as software--this is called "uploading". Now that your mind can be made into software, you can do interesting things: you can transmit your mind over vast distances using radio signals. You can "download" your brain into other bodies (called "morphs" or "sleeves").