In the newspaper, I saw a web address where one can listen to audio streams of recordings made 100 years ago and older.
http://www.cylinders.library.ucsb.edu/
I'm having a difficult time listening to the streams. Clicking on a link in Safari brings up the Downloads window, but when I attempt to open the .m3u file, it disappears. The site has a Featured Cylinder in the upper right corner, but when I click on the QuickTime player shown, it disappears and a portion of the big, blue Q appears in its place.

How do I correct this?

I'm listening to one as I type this. I get the same thing clicking on the stream. It just disappears. Clicking on the MP3 brings up the quicktime player in the next window. If you are only getting part of the Q, give it a chance. It may be just slow to load. You might also try option-clicking on the MP3 to try downloading it.

Saw that site listed also, it's very interesting stuff, and seems to work OK for me.

On my system (10.4.5/Safari 2.0.3), at bottom of the "Featured Cylinder" area at top right is a miniature QT player bar. Clicking the right triangle 'Play' icon starts playing the selection listed in the box. The other QT player controls operate normally. Looking at page source, if you have Quicktime plugin installed this is what should be happening.
Check Quicktime settings in System Preferences and see if you have 'Instant On' enabled under Streaming tab, and "Play Movies automatically" checked under the Browser tab. You may need to set Safari prefs to "open safe files after downloading" for this site.

Clicking the 'live stream' link in the "Cylinder Radio" starts playing the selection, but does so by launching iTunes and using it to play the m3u (streaming) file. In iTunes prefs, Advanced>General. try checking the Set button for "Use iTunes for internet Music Playback" and see what happens. The streams should show up in your iTunes Library as "continuous" in the Time column, and once accessed you can play them in iTunes anytime, just like other internet radio streams.

In the site's "browse" section, if I single click a "Download" link, the QT plugin activates and the MP3 file plays in that window. Right-click/Control-click offers the dropdown choice to download the linked file, as usual.

The site says they're getting heavy traffic since the publicity, so as Phyl mentioned, the load times may be longer than expected for these small files.