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

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!