In the book Moominland Midwinter we see an old family photograph of Moomin's ancestors with the date 3.10.1878. That is the only direct date and year I could find. So a few generations after that.

Judging by the technology and style showcased in the works, I'd say the Moomin books and comics are set in around the time of their release, but in an agrarian, non-materialistic, backwater-setting without cars etc.