About 6000 years ago.
But I don't think the exact details of when were at all important to Tolkien.
Tolkien had written multiple times that Middle-earth is located on our Earth.
He has described it as an imaginary period in earth's past, not only in The Lord of the Rings (see Prologue and Appendices), but also in several correspondence letters, estimating the end of the Third Age to about 6,000 years before his own time, and in N.W. Europe (Hobbiton for example was set in same latitude as Oxford), though at times he would also describe elements of the stories as a kind of "...secondary or sub-creational reality" or "Secondary belief" in replies to letters.
From Middle-earth on Wikipedia
scifi.stackexchange.com/users/35/Mark Rogers
I'm pretty sure that Tolkien intended his work to be a mythology of our ancient past.
It has been speculated that if LOTR ends at the beginning of the 4th age, then we may currently be in the 7th age.
That said, the length of an age is arbitrary and the events of the 3rd age do not fall anywhere in our own history.
You could think of the 7th age as all of known human history.
Anything older than that can only be considered myth.
scifi.stackexchange.com/users/148/TGnat
It was meant to be in this world, about 6,000 years ago.
In Letter #211 Tolkien addressed this specifically (the asterisk is to the footnote, reproduced below):
I hope the, evidently long but undefined, gap* in time between the Fall of Barad-dur and our Days is sufficient for 'litereary credibility', even for readers acquainted with what is known or surmised of 'pre-history'.
*
I imagine the gap to be about 6000 years : that is we are now at the end of the Fifth Age, if the Ages were of about the same length as S.A. and T.A.
But they have, I think quickened; and I imagine we are actually at the end of the Sixth Age, or in the Seventh.
scifi.stackexchange.com/users/4056/dlanod
I figured the stories took place pre-Genesis (in the Gap-Theory gap) like Robert E. Howard's Conan - Hyborian Age.
scifi.stackexchange.com/users/1654/Jeffrywith1e
Edit:
please note that the question explicitly states the time frame from 'the real world' and makes references to film props etc.
therefore what the author intended is not the question.
The questions is purely concerned with where in the course of real history the films would be fit in.
If you wish to downvote, please feel free but could you also leave a comment
so I know why?
Looking at the films Gondor is about 1450 and the shire is about 18c.
Please note this is based upon the full plate armour used by gondor's soldiers, orcs and elves (eg barbutes), the pole block tactics used by the orcs, the agricultural machines used in the shire and the crockery and cutlery used by hobbits etc.
scifi.stackexchange.com/users/6581/Stefan
C.S. Lewis' Space Trilogy and Tolkien's Lord of the Rings Trilogy was a collaboration, taking place in our universe.
In fact, C.S. Lewis referred to Numinor several times over the course of his book That Hideous Strength , which was set in post-WWII England.
From the Prelude he writes: Those who would like to learn further about Numinor and the True West must (alas!)
await the publication of much that still exists only in the [manuscript] of my friend, Professor J. R. R. Tolkien.
According to the antagonists in C.S.Lewis' book
That Hideous Strength , it would have been sometime before 110,000 years ago (before the beginning of the last glacial period ).
In discussing Merlin, Frost states: "What we have here," said Frost pointing to the sleeper, "is not, you see, something from the fifth century.
It is the last vestige, surviving into the fifth century, of something much more remote.
Something that comes down from long before the Great Disaster, even before primitive druidism; something that takes us back to Numinor, to pre-glacial periods.
"
Of course, these are the same folks that predicted that Merlin would join their side, and were subsequently devoured by their own future vivisection experiments; take what they say with a grain of salt. ;)
scifi.stackexchange.com/users/8590/S. Albano
Seeing how Lord of the Rings stems from Beowulf and the Nibelung saga as well as the Poetic Edda, 5th to 6th century would be a good guess.
The sword that was broken and remade, the cursed ring, invisibility, the dragon and his hoard, and the stolen goblet, and even many of the names (Gandalf, Balin, Durin, anyone?) stem from there.
scifi.stackexchange.com/users/952/dm.skt
I know that  The Lord of the Rings  is located in Middle-earth and by the looks of clothes and props, we're looking at well before the 18th century.  What's the best guess at the time period that  The Lord of the Rings  was set in?
In what time period does The Lord of the Rings take place in the real world?
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<the-lord-of-the-rings><tolkiens-legendarium>
scifi.stackexchange.com/questions/2237
scifi.stackexchange.com/users/630/benhowdle89
Lord of the Rings is supposed to be set in this world, approximately 6,000 years ago. Considering the films, some believe that Gondor is set in around 1450 and the Shire is approximately 18th Century. It is generally considered that the Lord of the Rings is set 6000 years ago. The costumes and lifestyle, on the other hand, appear to be 15th century for the orcs, elves and men, whereas the Shire seems to be in the 18th century.
The Lord of the Rings is considered to be set 6000 years ago by most people. Some say pre-Genesis. Costumes and lifetsyle appear to be 15 century for the men, orcs and elves and 18th century for the hobbits. Other believe 5th or 6th century because of references to Nordic Traditions like Beowulf and Nibelung.