Where I think it tends to fall down for a modern listener (assuming good recordings, which is not a good assumption) is that it mostly dates from a pre-recording period, so it tends to draw out all of its ideas because it expects the listener can't just rewind and listen to the last few minutes again, and this is the only such music they may hear this month; in modern times this results in a lot of pieces overstaying their welcome. Sometimes by a lot.
For instance, Beethoven's Fifth Symphony is a masterpiece. No question. But... the first movement has a repeat bar around the entire thing. Straight up. Not even a bit of variation in the last couple of bars as often happens. Now, of course, a good conductor and orchestra will play it with some variation even so, it won't literally be simply the exact same sound twice. But even so, it's not something a modern composer is likely to do. (Modern music does do a lot of repetition, but not on this this timescale. Trance a bit, but even then I'd say, it's different; modern trance artists know the modern music playback setups and handle it differently, even if I couldn't exactly describe it. Plus the 5th is definitely not "trance".)
I've sometimes fantasized about going into some of the classical oeuvre and editing it a bit for more modern sensibilities, by doing things like taking out the repeat, and maybe even incorporating some repetition with variation back into the piece, but generally shortening it. Or just generally toning out the repetition in some of the pieces, like, the Moonlight Sonata. Brilliant piece, masterpiece, no question. And the first movement is about 7 minutes to play. In modern times, it's unnecessary long and repetitive. I just listened to it here before posting, and it's not even like it's constantly mutating and changing or anything... it's just, repeating its ideas several times. For the time, great. No sarcasm. I mean that. Today, a bit tedious, because we don't need our repetition built right into our music, we add it with our playback technology if we want it.
I criticize neither classical music, nor modern listening practices in this. I merely observe a significant mismatch. Both have their reasons for what they do.
Also when a lot of people think "classical" what they actually pull up in their mental inventory is "chamber music", which is actually generally designed to be flat and unobtrusive and perhaps even, yes, a bit boring, because it's literally designed to be background music.
One option I like is to listen to music by artists who have clearly been classically trained, but are not producing classic/academic music. Even when such an artist is in the depths of the most electrotechno music possible, seemingly as distant from "classical" as you can get, I still find there's a depth that an non-classically trained artist has a very hard time reaching. (Sometimes they do, but it's rare. Though when they do it's awesome, because that's a truly unique musical experience.)
(In this post I use the slang meaning of "classical" to include all the time frames and almost all uses of dedicated "orchestral" music. I'm not even trying to be precise about "Baroque" or "Romance" or arguing about modern orchestral music.)