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Jonathan Harker's first entry in his diary reads, "Wednesday, 3 May 1893.
The current year is never cited in the Novel. The previous answer appears to be derived from one of the movie adaptations. It's a pretty good guess though. In Mina's jounal entry from August 1, the old sailor (Mr. Swales) relates that he's sat on a tombstone dated 1873 for, "...off an' on nigh twenty years past...".
Dracula (https://books.google.com/books?id=7NRDAQAAMAAJ) page 63.
In chapter 8, the night of august 11. is described to have a full moon, there was a full moon on that date in 1889. However, there was an almost-full moon on that date in 1892, and in 1897.
So, astronomically, those are the options for the setting of the story.
Book, chapter 8. Astronomy.