I greatly enjoyed the story and the idea that computers would gain awareness but eventually get bored of us, however I can't recall any specific hints about when the story is meant to have taken place. Any ideas?
As far as the precise year is concerned, there's no canon mention, nor has Spike Jonze mentioned a specific year in his interviews other than describing the film as "timeless" and set "in the near future".
The script describes the setting as;
"Slightly in the future, the city’s been developed even more with massive office, apartment and mall complexes. It’s a city designed for comfort and ease."
and for the record, there are no dates listed on any of the numerous computer screens seen at the main character's home, office or in the train station, nor does anyone say a year during the dialogue.
The only slight clue is a series of pictures on one of the computer screens entitled "Happy 50th anniversary". The first picture seems to be of an image quality (and with clothing) consistent with the mid 1970s which would indicate that the film is set no later than about 2025.