My understanding is the name Beryl is ponounced like the semi-precious stone of that name; rhymes Cheryl or sterile. Not a perfect match for "barrel", but closer if "barrel" is being used as a minor, unstressed word in a sentence. Closeness depends in part on the speaker's accent: a brit, for example, would pronouce more broadly the a in "barrel".

There is also the name Barrell (masc., yiddish), never common, but I've never heard of anyone having that name younger than may grandparents' generation.