Feeling like you might vomit; likely to cause vomiting; nausea-inducing.
Formed from 'barf' plus the informal '-y' suffix (from Old English), which turns nouns into adjectives. This is slang formation common in English for creating colloquial descriptions.
The '-y' suffix is one of English's most productive tools for casual adjectives—you can add it to almost anything and people understand immediately ('grumpy,' 'sleepy,' 'barfy')—which is why kids' language and informal speech are full of these formations.
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