The stardate is displayed in your logs. The dates tick up 1000 units per year, so it's very easy to work out a year based on the stardates. With a little more math you can even work out the month, day and time that the date represents down to the minute. The stardates used in STO take the current date and add 399 years as a base date for the stardate calculation.
Of course I heard from a dev that even though the stardates are in 2411, the storyline is still in 2409 for simplicity's sake.