I initially posted this in Linguistics, but wanted to get philosophers' opinions on this as well. (And someone over there is complaining that it's a philosophical rather than a linguistic question... who knew that meaning isn't the concern of linguistics?)
I think it's regarded as common knowledge in linguistics—since Chomsky—that there can be grammatical sentences that are nevertheless meaningless. The standard example trotted out for this is
Colourless green ideas sleep furiously.
Whether or not there can be meaningless but grammatical sentences, this doesn't seem to me like an example of one. To me, it simply seems meaningful, but false. It seems to me to wear its truth-conditions on its sleeve: for it to be true, there'd have to be ideas that are colourless, green, and sleep furiously. But these conditions aren't met: ideas aren't green, and since they don't sleep, they certainly can't sleep furiously. So the sentence just seems false—indeed, necessarily false, since it's impossible for something to be both green and colourless. But for a sentence to be false, it must be meaningful. So the sentence is meaningful.
Essentially the same question was raised in the Linguistics forum previously, although both the question and answer there seem a bit muddled, with the poster bringing in logic and Explosion when they seem beside the point. A similar question was also posted previously in this forum, but it's more specifically to do with Carnap on "Caesar is a prime number".
I found a similar discussion in Quine's Theories and Things, where he outlines a position similar to mine, in opposition to Ryle, Russell, and Carnap:
Many... predicates will be useless in application to attributes; thus it would be false, at best, to affirm, and useless, at best, to deny, that an attribute is pink or divisible by four. Ryle branded such predications category mistakes; he declares them meaningless and so did Russell in his theory of types. So did Carnap.
Over the years I have represented a minority of philosophers who preferred the opposite line: we can simplify grammar and logic by minimizing the number of our grammatical categories and maximizing their size. Instead of agreeing with Carnap that it is meaningless to say 'This stone is thinking about Vienna', and with Russell that it is meaningless to say 'Quadruplicity drinks procrastination', we can accommodate these sentences as meaningful and trivially false. Stones simply never think, as it happens, and quadruplicity never drinks. (Theories and Things, 1981, p. 110)
Quine's position seems to me to be quite right, and, applied to "Colourless green ideas...", to lead to my above conclusion. However, Quine says that his position was a minority among philosophers during his time; is it still a minority position today? In the Carnap on "Caesar..." thread, philosopher Stephen K. McLeod is quoted as saying something similar about that issue, so it seems like there's still at least one contemporary defender of this view.