That's still ~500x faster than any man-made object has ever moved.

I don't think there's any reason to think that we've reached any kind of serious limit at the moment, or that a technical civilization, say, 10,000 years older wouldn't be much more capable. There's certainly no fundamental physics in the way of that sort of velocity.

Well, as long as we're fantasizing, make the speed 0.00001c and a billion years.

It's in the same order of magnitude as the Project Orion speed estimates, based on present-day technology - it ought to be possible if the political will was there.

While true, Isn't this more an "engineering problem" barrier (like the sound barrier) instead of a "theoretically impossible" barrier?

There may be an engineering barrier below c when you account for fuel efficiency. Once you start requiring large volumes of nearly impossibly efficient (thrust-weight) fuels, then you will be stuck, even if you aren't anywhere near the speed of light.

It depends how much material you are transporting. If you are sending a few photons, the speed of light it no barrier at all. But if you are sending a crew of humans...