The short answer: I think it's very unlikely with today's technology. But there are two ways to interpret the question. Do you mean a machine that can be mothballed today and persuaded to work again in 3000 years time? Or a machine that can remain in service for 3000 years, with regular maintenance?
A mothballed machine is in a different state to that it was designed for. Lubricants will drain out from between moving parts. Slow chemical changes will take place. Classic car enthusiasts occasionally find a car that was abandoned in a barn or suchlike decades ago. Although it has been protected from the elements, it will take a lot of work if it is to be made roadworthy again. The battery, obviously, will be dead. A lead-acid battery discharges itself in less than a year and then sulphates, making it unable to take a recharge. The rubber components (tyres, belts, wipers) will have rotted. Rubber degrades in air. In general it's not regarded as safe within a decade and is no longer rubbery but hard and brittle within three. Less obviously, not only will the bearings no longer be lubricated, but they may well have seized. Slow chemistry can cause metals in contact to weld together, or lubricants to oxidize and polymerize into glues. The engine will need careful dismantling, cleaning, and reassembly, with some replacements.
This, after fifty years. After two hundred, will it still be possible to obtain a lead-acid battery? Or the tightly specified fully-synthetic oil that most modern cars require? Or a compatible tapered-roller wheel bearing? Or even the gasoline to make it go? By then we expect the internal combustion engine to be rather quaint history, and everything to be running on renewable electricity. (That, or society to have collapsed back to the mediaeval level or worse, due to global warming and environmental degradation, and the old car will be beaten into a ploughshare or something).
After three thousand years? Sorry, I think any but the most simple of mechanical devices will have by then become inoperable past the point of being restored. Any interest will be archaeological, or if society has collapsed, possibly the inspiration for an engineer who can puzzle out what the old heap was for and then build a new something.
(I'm assuming that it's been kept in a very stable and dry environment. If it's made of iron in a damp environment, it may well be a heap of rust rather than a recognisable mechanism! )
Now, what about a continuously maintained mechanism? I believe that the oldest continuously working devices in the world are clocks and windmills. The oldest clock is believed to be in Salisbury Cathedral, dated 1384. There is some uncertainty in that records from that far back are not good and the design of church clocks did not change significantly for the next century at least. But if it's not this one, there are plenty of other candidates from the 1400s. At least one will be an original. The astronomical clock in Prague is well-documented to have been built in 1420. (It was seriously damaged during WW2 and restored ... is it still the original)?
In Iran there are some windmills, still in use, that are in buildings probably constructed during the 6th century. The simple mechanisms are built of wood, leather and suchlike. It seems most unlikely that any components made of these materials and used daily can be 1500 years old. So at what point does a regularly maintained machine cease to be original? When the design is altered? When the last original component is replaced? Later?
(At the atomic level there is almost certainly nothing at all in you that was there when you are born. Most of the atoms of living tissue are replaced within six weeks of now. But we remain the same people).
Many mechanisms last a century (without major replacements) and are working just fine when they are scrapped. They've become obsolete. There were many steam locomotives built in the 1850s that were in service in the 1950s. Then along came diesels, and electrification. A few are still working, courtesy of railway preservation societies. They're just no longer mainstays of the transportation industry. For how many more centuries will anyone be interested in maintaining a steam engine in working condition? Ditto, a windmill?
And finally, there's the life of the society to consider. 3000 years is a long time. Few if any human societies have lasted that long. None, without massive social upheavals involving considerable destruction. Climate change happens on a shorter timescale.
It's just about possible that a clock will survive 3000 years given occasional maintenance. It's something that is has been good enough since the 1400s, will stay good enough as long as people remain big-brained apes. A clock moves and wears out slowly and an old one does not need any technology higher than the mediaeval to maintain it. But will the church or other building that it is part of last that long? The omens are not good. The Pantheon in Rome is probably the oldest building continuously maintained and used, and it's about 2000 years old. Earthquakes, fires and wars will continue to take their toll.
Which is why I don't think it will make much difference if we can engineer (say) clock components that will last three thousand years of use without needing replacements. It's what's around the clock that won't last the distance. In many places, not even the geology will. What's left of Roman London is now ten to twenty feet underground.