You can use audio player from your own Google reader, GMail, voice and use the embed code @leo provided, replacing the link to the .swf file!
Update: Google reader is retired, better save the .swf player file from gmail or voice, host on your own i.e dropbox, Google Drive, etc
Answer to your question is no you can't host audio on blogger servers, earlier they allowed it when FTP blogs were allowed, but now only video upload is available, which some use for hosting audio using a smaller truncated youtube player!
there are two things:
a. host audio;
b. host audio player;
You can use combinations that fit you in security, privacy and convenience
- Host your audio someplace safe (means public access to the file remains at least as long as the service remains) such as amazon 5 GB, Google Music and use audio player from Google to embed it.
Update: Amazon's cloud drive doesn't offer public links.
Host your audio on services which offer their own players soundcloud which has 2 hour limit on uploads, my favorites: divshare 2 gb (no time based limits) and spreaker which offers 20 hours storage! there are countless others.
Use external hosts like dropbox, google drive, box for both player and audio, use google audio player.
If you don't want flash players, use these jquery HTML5 players >> mediaelements.js, soundmanager, jplayer.
Update: Dropbox now allows public links anywhere in your folder structure, just generate a sharing link, remove the subdomain part to dl, i.e replace www by dl
You can host mp3 in Google drive, change permissions to read-only to anyone with link, it is not a direct link but it is to preview the file, you can download the file and then immediately cancel it, copy the download link in your downloads tab, remove the session id or timeout variables after the (?), use the link with the embed code by Leo!