Does Your System Sound Like the Real Thing?

I would say that with certain source material, yes. Scale is the thing that keeps me from fully believing on many occasions. My vinyl is better at scale than my digital. But it all seems to boil down to the quality of the recording, whether analog or digital. There are times though when it seems quite real.

"If the answer is no, do you even think it's possible?"

An anal response to your specific question, I don't think a speaker exists that could pull it off. Because there is some compromise at any price point, they always leave a hint of themselves. One could be tricked with given sounds, but I believe one going to be limited by either distortion or dynamics in the overall delivery. That is what I have noticed, anyways, although I have not heard the best of horns. And scale is definitely a problem. I use speakers because they are an easy target, but I couldn't imagine a CDP pulling it off either. Maybe vinyl, maybe not, but that's a loose thought. One would get closer to the "real thing" with an original tape, but exploring how far from THE performance playback could stray before sounding distinguishable would be interesting.

"And if you do think it's possible, how far are you willing to go?"

The upshot is that I think one would have to be insane not to find a satisfactory system, minus the money issue (maybe), and some just intrinsically enjoy finding the limit of hifi.

Some live performances sound pretty bad... some recordings have excellent sound. In a very good large hall if you're 100' away from the musicians you'll miss alot of nuances that are easily picked up in a good nearfield playback environment. I probably haven't heard anything near the best possible playback system, (there may not be one that does everything the best), but so far the best live stuff knocks the stuffin's outta the best recorded stuff w/ out breaking a sweat. The two things are just physically (and psychologically) very different from each other. Then there's the can of worms regarding the perception of musical or artistic merit vs. recording quality (who wants an exquisitely detailed rendering of something that hits the gag reflex?). There are lots of acts that might make you head for the door on their most sonically perfect night. Then sometimes when you least expect it you could end up really liking some stuff that you used to hate.

No and No.

your system is indistinguishable from live performances

No way.
One system might fool you -- but that's a musicality issue. The other is higher rez -- but that's "true to SOURCE" i.e. the recording, NOT the musical event.

IMO that's all we can achieve -- and it's not bad after all: either a "musical" sound reproduction OR a "true to source" reproduction. Whatever tickles our collective fancy.

So,

If the answer is no, do you even think it's possible

Of course not.
Not only are recording techniques & storage media a limitation; the speakers & spkrs-room interaction are even more so, as Ohlala notes above.

BTW, and similarly, a portrait is NOT the actual person, is it? It's an artist's interpretation and rendition of that person. (I'm putting sound reproduction on a high pedestal here:))
Cheers