Learn Hardcore German: Das Regal ist einen Meter breit.

Because German often uses the accusative for measurements with adjectives like breit, lang, hoch, tief, schwer, alt. This is called the “adverbial accusative of measure” (Akkusativ der Maßangabe). Meter is masculine (der Meter), so the accusative singular is einen Meter. Hence: Das Regal ist einen Meter breit.

But doesn’t sein (“to be”) take the nominative? How can there be an accusative here?

You’re right that predicates after sein are normally nominative (e.g., Das ist ein Regal). But here, einen Meter is not the predicate noun of ist. It’s an adverbial measure that belongs with the adjective breit. So the case rule for sein doesn’t apply; the adverbial accusative does.

Is Das Regal ist ein Meter breit also correct?

You may hear or see ein Meter breit, but the standard and most natural form is the accusative einen Meter breit. In careful or formal German, prefer einen Meter. Regional or casual usage may accept the nominative, but it will sound off to many speakers.

Could I say Das Regal ist einem Meter breit?

No. Einem is dative. The construction needs the accusative of measure: einen Meter.