DropBox actually would let you access the songs when you are using your phone, as well as provide access to the songs for other devices such as your PC, or make them accessible over the web to other people (they can access publicly shared files using a webbrowser).
Maybe rephrase your first post a bit to make it a bit more clear?
From your post, I get that you have a collection of music that is not available on streaming sites, so you want to be able to play them on your phone and you want to stream them, which basically means you want to make the music you have accessible for other devices or other people.
I assume you don't have any problems in making the files play on your phone. As long as they are in the correct format, the music player will play them for you.
For the streaming, DropBox syncs a folder from your phone onto a folder on the internet, so whatever files you have in this folder will be on the internet, even if your phone is off. Then, it is just a matter of accessing your DropBox share on a different device, and you can play the music files directly from there.
There is a topic on this in regards of video streaming on these forums already, and it will work just the same for music.
If you want to actually "DJ" with the songs, or provide something like Pandora with these music files, than you will have to set up something like a shoutcast server of your own.
http://www.droidforums.net/forum/droid-audio-video/59636-streaming-video-android-dropbox.html