Get one of these little wireless routers and attach a flash drive loaded with audio files.

Spotify lets you download music so you can listen to it offline. This is only available with paid accounts, but it’s the easiest solution. It means you can listen to the same music, using the same streaming software, offline as well as online.

I am not sure of your question but if it is to rip a stream to hard drive and have files named to the current playing song, Streamripper is a great command line tool. There are also GUI front-ends for it. Check the main page. It only works with Shoutcast streams but there are many free internet radio stations using Shoutcast. It does not work with last.fm streams though if that is what you are looking for.

Google Play Music allows you to pin your music (making available offline) on all levels of account. The free version allows you to upload up to about 10,000 songs, manage playlists etc. That you can stream and cast via your phone. Pinned items (songs, albums etc.) are available offline as you describe.