My own objections to the meaningfulness of paradoxical statements are as follows. First, can a paradoxical statement apply, that is, be instantiated, as describing a physical object or event? Paraconsistent logicians seem to agree that it cannot. Second, cannot we resolve the apparent problem of paradoxical statements by arguing that we misunderstand these statements, even though we think we do understand them?
I'll start with the second claim. Here are the counterarguments to the meaningfulness of paradoxical statements, abbreviated; they're expanded below.
2a. Paradoxical statements are nonsense.
i.e., saying "This statement is false" is similar to saying "The swam cloud in climbed alligator the".
2b. Paradoxical statements are not evaluable as propositions.
i.e., saying "This statement is false" is actually like a question or command, which is a statement that has no truth value; but we're confused by it because it appears to be a proposition.
2c. Paradoxical statements only seem interesting because their wording tricks us into applying logic inconsistently and serially; we're "solving" them incorrectly.
i.e., claiming that "This statement is false" has more than one truth value results from incorrectly serially applying values ("if it's false then it's true, so it's false, so it's true"); computers serially manipulate values as we are doing when we think about this, but this is wrong; here we're looking for a value associated with this statement that is always the case, all the time, and was the value before you started thinking about it and is not path-dependent.
2a. Paradoxical statements are nonsense. When we say something like "This statement is false" or any of the other paradoxes allowed by recursion, an easy if boring resolution is that we're actually just babbling, the same as if we just said "The swam cloud in climbed alligator the". We have piled words together that themselves have meanings, but in a way that precludes meaning. The objection to this is that these phrases seem to mean something, that they are grammatical and we understand them - and in some ways we certainly do - so there must be some meaning to them. And a simple objection to this argument is that we feel, intuitively, that we have understood things when we clearly don't understand them all the time. That intuition is not grounds for saying something is other than nonsense, if there is otherwise a clear problem with it.
The claim that the statement is grammatical and therefore must be true or false is a non-starter (see Chomsky). "Beethoven's fifth symphony is purple" is nonsensical but completely grammatical, and the fifth is neither purple, or non-purple; given the specific nature of the described, the descriptor can't apply. Possibly there is a Russell-like solution here that states when a statement is recursive, we're just not realizing that parts of it are automatically in different classes of objects to which certain properties cannot apply; i.e. a statement cannot refer to its own accuracy, anymore than flavor can refer to a symphony.
2b. Paradoxical statements are not evaluable as propositions.
Truth and falsehood only apply to propositions. They do not apply to objects ("The sandwich is false" is meaningless, regardless how bad of a sandwich it might be; this statement is like the purple symphony above.) There is a longer discussion here about the purpose of language in humans, the fact that it's really a way for us to cause each other to behave differently, the likely non-coincidence that in almost every language the command is the most basic form of the verb; yet commands can be neither true or false. In Priest's dialogue he points out that it's nonsensical to talk about physical objects' being true or false, and that in fact this is also true of certain classes of statements, though he focuses on questions - and in fact questions can be thought of as commands to supply information that produces a true proposition; i.e., asking someone "What is the capital of Assyria" has the same effect as commanding them "Tell me the capital of Assyria" or (more completely) "I don't know the capital of Assyria, and I want you to give me information that, when filling the blank in the sentence '[blank] is the capital of Assyria' produces a true proposition."*
Therefore, it may be that paradoxical statements are commands or like commands, in that they are utterances that do not have a truth value; we're merely confused by their appearance of being a declarative statement which therefore must have a truth value. Although this makes the problem of paradoxical statements go away, it remains to be determined what kind of utterance paradoxical statements; they don't seem explicable in terms of commands.
2c. Paradoxical statements only seem interesting because their wording tricks us into applying logic piecemeal, inconsistently and serially. They do mean something, but we're "solving" them incorrectly. That the meaning of these statements appears to oscillate as we work through the steps is our signal that we're "doing it wrong". A logical value associated with the statement just is, and is timeless; that value cannot have anything to do with the steps or path we take to get there, but the structure of language forces us to move through those steps serially and give us a nonsensical result. Of course the assertion about the timeless, substrate-independent nature of logic in particular is assaulted by some mathematicians who think it evolves over time and is even dependent on discovery by thinking beings.
The most useless machine. It is a physical object and thus described in serial, caused states. There is no paradox that it can be both on and off.
Now recall the first objection: that paradox can only result from statements (which can describe anything, including themselves) and as such cannot point to any object in the physical world. Our own thoughts and our sense of understanding are, of course, themselves object/events in the physical world; that is, they are certain states in our brains. Therefore, if you agree that paradoxical statements do not point to anything in the material world, to be consistent, you must also agree that we cannot really understand them - because our understanding of paradoxical statements is itself a physical thing. To do otherwise is to reject materialism. The line of counterattack here is to what degree our thoughts have the same logical properties as the things to which they refer.
*Priest points out how interesting it is that paraconsistent logic re-emerged as an area of interest in the late 19th century when the foundations of mathematics were being investigated. If this is not a mistake or coincidence, this is fascinating, and I argue says more about how we construct propositions (in formal systems or plain language) and therefore more about our own neurology as animals, than anything in the rest of the world.
*(Questions lacking interrogative pronouns are actually full propositions which invite the hearer to assign a truth value. "Are you hungry?" "No." is really "I provisionally state 'You are hungry' and declare my ignorance of the truth of this statement, as well as my desire to correct said ignorance, with the tone of my voice and/or a particle." "False.")