The American Dream of today is probably more similar to what it was in the 1920s than any time since the Great Depression of the 1930s.
The American Dream in The Great Gatsby is depicted as the opportunity for any person to start over in a fresh new place and realize their dreams. These dreams include achieving wealth and success, as well as finding love and fulfillment. They involve being able to set back the clock as if the past could be erased and everything started over on a better footing. This dream fills the hearts of the sailors who first see America, which seems untainted to them by corrupt civilization, and it fills Gatsby's heart with the desire to erase the five years he and Daisy have been separated.
The American Dream is most like this today because we have gone back to a moment when entrenched wealth makes it more difficult than it has been at any time since the 1920s for a poor young man like Gatsby to get ahead or get access to upper-class society. Like then, illegal means, such as those which Gatsby pursues, can feel today as if they are one of the few avenues open. Second, we live in a moment of intense, nostalgic desire, at least for a large portion of the population, to turn the clock back to an imagined, Edenic past.