What are the characteristics of Hifi geared towards good reproduction of classical music?

There are several things which have contributed to classical music in my system. One is the ability of the system to produce not just a wide, but a deep and articulate soundstage. Instrument positions should occupy distinct positions, as they would in a live concert. This is accomplished by a number of factors -- speaker position, room acoustics (including treatments such as diffusion), and a good source and DAC. (Good recordings are, of course, a must.)

The other element which has aided classical music reproduction for me has been having multiple subwoofers. This helps create both the soundstage of a large orchestra as well as the low sounds produced by, say, string bass, timpani, and some of the woodwinds and horns.

Finally, I have been impressed by the ability of some systems -- mine, to a limited degree -- to produce string sections which sound like many individual instruments playing together rather than just a mass of strings. Ribbon or planar tweeters can do this especially well.

For me, accurate orchestral string tone and absence of electronic artifacts in vocals are the two most important factors for reproduction of classical music. Imaging and three dimensionality can put me on Cloud Nine, as well. Dynamics and punch are important but it certainly isn't the last word for me. In any case, I've recently found a whole lot of the aspects I love in my NOLA Boxers, which feature soft dome tweeters.

Dynamic headroom is very important handling all classical well including symphonies and other large scale works. If the amp runs out of gas driving the speakers properly during crescendos the result can be a muddled mess. Likewise the speakers need to handle those dynamic crescendos well without undue compression so even very good quality smaller speakers may be handicapped except perhaps in a smaller room. Room size and how far away from speakers you are matters. Crescendos should be able to go at least into the 90db spl range for any sense of realism. Use a sound meter or sound meter app on a smartphone to measure. Louder is better (live symphonies will frequently exceed 100db by a good margin) but once you get past mid 80s db spl regularly you are into levels that can be harmful to hearing. Once the amp and speakers are up to the task then you might play with tweaking the rest to personal taste.  If the rig can handle the large scale works the easier smaller scale ones will come naturally perhaps with just a tad of tweaking in many cases.    Room acoustics are always a big factor and varies case by case. 

The same characteristics of gear for any other genre.

Timbres, tonality, dynamic range, imaging, soundstage, cohesiveness top-to-bottom and clarity.

A room and system either sound varying degrees of  good or not, regardless of the genre.

@tomcarr

Good answer.

Timbral resolution (instrument identification) always must come up very near the top for both classical and jazz.

Unfortunately far far too many systems fail to deliver.

Some good attributes have been brought up. But I think to get them the first step is speaker selection. This will drive all other choices. I would assume that over the long run you will need to spend ~35% on speakers (highly variable depending on your route… but a good number to start with. 
 

You want the speakers to emotionally connect with you. Personally I think these days that dynamic speakers are really good way to go since you will not have to battle very difficult electronics requirements and more finicky placements. Also, if you look at the market place, these at all levels are the most common for a reason (let’s not get carried into a battle of what that reason is). 
 

You want the speakers to fit your room… I mean not overwhelm it or not be dynamic or powerful enough. 
 

Once you find the speakers, then you can start looking for appropriate electronics to drive it. You can see my systems under my ID. I listen to classical, jazz, world, blues, and rock most of the time.

Sincere thanks to all those posted - fun and educational...

Merry Christmas everyone!