The underlying music for a large fraction of classical music is public domain now (but not all...classical music never stopped being composed).

But copyright can also cover the performance itself, not just the underlying music. So even though Beethoven's 9th symphony has long been public domain, for example, the Telarc recording of it by Christoph von Dohnanyi and the Celeveland Orchestra & Chorus is not. If you want to do something with that specific recording, you need permission.

Also, a particular arrangement might be copyrighted. Consider Mussorgsky's "Pictures at an Exhibition". That was composed as a piano piece in 1874, and long ago became public domain.

Since then, numerous other composers have arranged it for orchestra, with the most well known arrangement probably being Maurice Ravel's in 1922. There have been dozens of other orchestrations including several this century, and also numerous arrangements for other types of groups and combinations of instruments and solo instruments.

A given recording will only be public domain if the underlying work, the arrangement, and the performance are all public domain.

Performance aside, plenty of classical works aren't out of copyright. My favourite composer, Herbert Howells, died in 1983; his works won't be out of copyright until 2053.

Recordings are often recent.