Trying to pinpoint the exact year — or even decade — that ASOUE exists in is almost as hazy and unpredictable as the series itself . . . which may be the point.
The show, based off the books by Lemony Snicket, features technology that points to the 20th century. First, there's the use of rotary phones and the obvious lack of cell phones (which would really be a game changer). The characters use typewriters and are seen sending telegrams. In The Hostile Hospital: Part I, the hospital has no electronic records. Count Olaf's car is straight out of the 1970s, and at the beginning of the second season, the children arrive at Prufrock Preparatory School, where their new vice principal introduces them to an "advanced" computer . . . which is neither advanced nor what we would today consider to be a computer. These clues would put the series around the 1980s.
However, the children seem to be constantly clad in somewhat modern clothing (even though Mr. Poe's assistant, Jacquelyn, gives off major 1940s vibes with that lipstick and those curls), and modern technology is referenced several times, like in the second episode of the first season when Count Olaf says he bought an hourglass "online." And later on in the series when Olaf claims he prefers "streaming television in the comfort of [his] own home" — an obvious and delightful Netflix reference.
With the show's conflicting technological advancements and lack of current-event references like wars or presidents, it appears to be set in a time of its own — a warped universe separate from our own reality and historical timeline. Which is probably for the best . . . because unibrows have never been in.