From: Bruce O.

"Beowulf: It is clear that the poem as we now have it wasn't sung to a fixed tune as we would call it today. There are anywhere from 4 to 8 stressed syllables per line."

Well I'd guess that if you can't sing a sing with odd length lines, that makes the Psalms impossible as well as most of Paul Simon's songs!
Of course you can sing Beowulf, as long as you can get close to pronouncing the Anglo Saxon as if it was a living language and not an invocation to the Dark Gods of Hyperboria.
I know you can sing Beowulf, because I have done it- an extract from Chapter 11, not all of it- that would make Wagner's Ring cycle a three minute single by comparison!
There are academics who talk about the tune for Beowulf, mentions have been found of "up" and "down" phrases, possibly meaning imperfect and perfect cadences.   
Listen to the version on my website,
http://www.darowyn.co.uk
then tell me it can't be sung!
Cheers
Dave