Is it legal to offer a web service were people upload their music (that I don't know if it's stolen) and listen to it on?

Streaming is a dangerous game, particularly when you know, or should know, that most users and most of the streaming is illegal. Here's a key question: Is this website only for people to listen to their own music, or does it let you, for example, listen to the music belonging to a friend? The former can be relatively safe, legally-speaking, and the later not. Basically, you need to have a legal consult to hash out these details.

This is area that is fought with peril. YouTube operates on a similar model however, they have been sued by the music and film industry (and won) many, many times. Therefore, I believe that current law would allow you do do that which you propose, there are many safeguards that would be necessary to reduce the likelihood of being sued and if you are sued, so that you can prevail.

This is too complex to discuss on this site and I would also need to know all the details before I could even begin. Just like Myspace did back when it was a real functioning music site where every indie music artist uploaded their songs, you have an affirmative obligation to take steps to make sure the material is not stolen. Since starting such a website will cost a great deal of money, part of your budget should be for legal help.

You may upload and download your own original compositions and your performances of your compositions. You may also upload and download compositions and performances that were created or recorded prior to approximately 1923. Other than that you must get the permission of the copyright owner(s) for each piece (or their designated agent such as ASCAP or BMI) and pay the licensing fee for each piece which involves counting how many people listen to each piece each time it is played.

No, it is not legal to create a web site where people can upload their music if you do not know if they have legally obtained it. It is illegal for people to share music if they have not legally obtained the right to share it. If you create a web site on which they can upload music that has not been legally obtained, you will also be liable for violating the law.