At least several millenia
The series is clearly set at least several centuries after the modern period, since the Stalkers were destroyed (and presumably created) "hundreds of years ago."
"I thought they all died hundreds of years ago! I thought they were
all destroyed in battles, or went mad and tore themselves apart..."
Of course, we can do better than that.
The series is set around a millenium after the beginning of the Traction Era.
25th April 1027 TE
—A Darkling Plain
This is supported by Ermene Khan's thoughts on the Shield Wall:
For a thousand years his family had helped to man the Shield-Wall, and
he seemed dazed by the news that all his guns and rockets were
We further know that it must have been at least 1400 years after the 21st century,
Two black-robed Guildsmen hurried past, and Tom heard the reedy voice of old Dr Arkengarth whine, "Vibrations! Vibrations! It's playing merry hell with my 35th Century ceramics..
."He waited until they had vanished around a bend in the corridor,
then slipped quickly out and down the nearest stairway. He cut through
the 21st Century gallery, past the big plastic statues of Pluto and
Mickey, animal-headed gods of lost America.
That makes the modern period at least 1400 years before the time of the series. The dating of the ceramics is probably pretty accurate, since Pluto and Mickey were (fairly) correctly placed in the 21st Century gallery. Indeed, it seems likely that the 35th Century was before the Traction Era, since the use of TE to represent the Traction Era seems to be standard. This would place the time of the series at no less than 2400 years after the modern period (21st Century).
This is consistent with Hester and Pomeroy's comments, which indicate that the modern period was at least several millenia before the time of the series.
Hester says of satellites:
"Except for the ones that aren't really stars at all. Some of the
really bright ones are mechanical moons that the Ancients put up into
orbit thousands of years ago, still circling and circling the poor old
Similarly, Pomeroy says, speaking of St. Paul's Cathedral:
Thousands of years old, that cathedral, and they go and turn it into
a... into whatever they've turned it into, without so much as a
St. Paul's Cathedral was constructed starting in 1675. As such, we can suppose that the series is set no more than a few thousand years after the modern era, since even a stone building would degrade heavily after a while.
That's about the best we can do.