You are legally allowed to play the music in your restaurant because you are not making any money from the music you play, as a radio station does. When a radio station plays music, sponsors pay them for commercials because of the amount of people listening to the radio station for the music, so the station is indirectly paid by playing the music. In a restaurant, however, you can play even the radio and it doesn't matter because the radio station is paying the royalties for the music, you don't need to. And since the music is public domain, you have the freedom to play it in your restaurant.

Be careful! It's 3:30 a.m. and I don't have citations to hand, but would bet heavily that --use of music as background in a commercial setting is a commercial use, even if there is no direct advertising or charge made for listening; --while the Ninth may be in the public domain, a particular recording of it is probably the property of the artist(s); I'm sure it would be illegal for you to record the performance in the hall, or to use their recording other than for your private enjoyment. Note that if this is a real world question, not just hypothetical, you should CONSULT AN ATTORNEY. Any information acquired through a site such as this should be regarded as For Amusement Only. With that in mind, you might check whether ASCAP (The American Society of Composers, Authors, and Publishers) and BMI (Broadcast Music Incorporated) have web sites that might supply preliminary answers. (They're the two main US organizations that look after performance and publishing rights.) And another caveat: Beethoven is a very dubious choice for background music. He's so architectural, so logical, that sooner or later everybody ends up listening to the exclusion of whatever they were doing to begin with, like eating and conversing. I've seen it happen at parties. And if the second movement is starting just as I'm beginning dessert, I'm still not going to get up and walk out before the end, killing your turnover.

You should be aware that while the music itself is in the public domain, the recorded performance is not. The performance is almost certainly copyrighted by someone, probably the studio that issued the CD. That said, I do not believe there are any restrictions on playing such music in a restaurant as background music. Background music is frequently provided in restaurants and stores. If you are really in doubt, you could simply contact the copyright holder by e-mail for information on the matter.