There is no size limit as long as it is 'an object', and there is precedent for 'the hull' of a vessel to be treated as an object
Much of the answer to your question turns on what is a 'flying Illithid raiding skiff'. My cursory internet searches have not found such a vessel defined in 5e rules; most hits were to Illithid nautiloids and githyanki Astral skiffs. If the skiff is your personal invention for your campaign, you will need to decide how this answer applies to it.
Heat Metal may be cast on:
a manufactured metal object, such as a metal weapon or a suit of heavy or medium metal armor, that you can see within range.
You say that the skiff was manufactured and metal, so the only question is whether or not it is an object.
As it turns out, 'What is an object' is not an easy question in 5e rules. What is not an object is pretty clear - the skiff is not a creature, or a spell effect, or a dungeon hazard, so it is definitely 'object-class'. But is it an object, just one object for the purpose of the spell, or is it a collection of many objects? You ask whether it is too large to be considered an object - but size does not appear to be the determining factor. Rather, there seems to be a limit to complexity: is it composed of 'many' other objects? Unfortunately, while the rules suggest this as the criterion, they don't provide a rigorous definition of how many is 'many'.
The DMG says:
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
We are told an object is discrete and inanimate. "Inanimate" means that it is not a creature, and "discrete" means that it can easily be recognized as separate from the environment and other objects, and your skiff easily fits these conditions.
We are told that an object is an item, "like" a window, door, sword, book, table, chair, or stone. None of the examples explicitly reference size ('a stone', for example, can be very large), although all the examples are within the range of things that a PC can interact with. Many of the examples are composed of component parts; e.g. a sword has at least three parts (blade, crosspiece, pommel) that are joined together by a smith. While the smith was handling these items they were each separate objects, but once joined together they became one object, the sword. Similarly, a book was at least multiple sheets of parchment, many fibre cords or leather thongs, and two separate wooden boards all combined into a single item.
Consider a PC holding a book which, we are told, is an object. Suppose another PC tears out a page from the book and moves away; they are now holding a sheet of parchment which is itself an item and object. The book is no less an object because it is now missing the sheet, and it is still an object even though it might be composed of hundreds of these sheets. Thus even the number of parts does not seem to determine when something is no longer an object itself but must be treated as a collection of many objects.
I have to conclude that an object is some thing with which a game entity interacts. This is a functional definition rather than a categorical or linguistic one.
PC1: "I hold the book while I ritually cast my spell.
PC2: "I grab the book PC1 is holding, rip out a page, and flee."
PC3: "I shoot a firebolt at the page PC2 is holding."
I conclude that an object exists at the scale of a game interaction, at the time that interaction occurs.
The DMG definition of an object ends with it being "not a building or a vehicle that is composed of many other objects."
These two things, buildings and vehicles, certainly seem larger than the other examples of an object we are given. And they might be more complex, such that one could interpret this passage as saying 'buildings and vehicles, since they are composed of other objects, are therefor not objects themselves.' That was, in fact, the way I read this for quite some time. I am indebted to Journer for pointing out that the phrase uses 'that', not 'which'. The actual meaning of this passage in the DMG is: buildings and vehicles, when they are composed of many other objects, are thus not objects themselves. That is, a building or a vehicle might be an object or it might not; they are objects when they are simple enough, but not objects when they are composed of many other objects.
For example, a cart is a vehicle, and it is also listed in the DMG as a "large object", and it is composed of what were once many separate objects. On the other hand, something like the Great Canoe is a vehicle 63 feet long and had a crew of forty people - and was carved from a single tree as a single-piece hull. Neither size itself nor complexity or number of parts themselves gives us a definition of 'an object'. Again, I am forced to conclude that what the game is going for here is not actual complexity but functional complexity - does the vehicle or building contain parts with which the PC's will interact, so that they should be considered objects in their own right?
PC's most typically interact with objects by using them ("Interacting with Objects around you" sidebar on PHB 190) or attempting to damage them ("Statistics for Objects" DMG 246), but there are many more obscure possibilities, such as those of a 17th level Knowledge Domain Cleric performing an Object Reading to "see visions of the object’s previous owner" (PHB 59). Thus whether or not something is an object itself, or is merely a part of another object, or is something composed of other objects, depends on the context in which the PC's are interacting with it.
For example, if a party was attempting to get through a locked door quickly, the barbarian might attack the door itself, in which case the entire door would be considered an object. But if they were trying to get through the door quietly, then the rogue might work at the lock while the artificer tried to disassemble the hinges. In this case the door would not be an object, but a collection of other objects, and the lock and hinges would be the objects with which the PC's were interacting.
Hopefully you can see where I am going with this; was the entire skiff of the size with which your PC's could interact at the time of the interaction, was it more appropriate to treat its component parts (like the hull) as objects in their own right, or would the spell interact with something even smaller, like one plate of the hull?
Turning to the ship rules in Appendix A of Ghosts of Saltmarsh, we find that:
A ship is composed of different components, each of which comprises multiple objects:
Hull. A ship’s hull is its basic frame, on which the other components are mounted.
Control. A control component is used to steer a ship.
Movement. A movement component is the element of the ship that enables it to move, such as a set of sails or oars, and has a specific speed.
Weapon. A ship capable of being used in combat has one or more weapon components, each of which is operated separately.
These components are both "used" by the crew and have their own AC and hit points should something attempt to damage them. Even the smallest and simplest watercraft listed as an example, the rowboat, has two different multi-item components: the Hull and the Oars. If your Illithid Skiff is of comparable complexity, it is likely that you cannot treat the entire skiff as an object for the purposes of heat metal; rather, you would need to select one component of the ship for the spell to affect. On the other hand, the largest vessel described by Ghosts of Saltmarsh is the galley, which has 130 feet of carlin (fore-aft length) and whose Hull is most definitely composed of many, many planks (equivalent to aircraft carrier plates in your question). Despite the gargantuan size and large number of component parts, the entire galley Hull is treated as one object within the ship rules. It seems within the intent of the rules for a single spell to be able to affect the entire Hull of a vessel, regardless of size.
For the specific context of heat metal, consider that one of the examples given in the spell description of 'a manufactured metal object' is a 'suit of heavy metal armor'. While typically sized for a humanoid, this could be as small as the plate mail of your Fairy PC or as large as your Elephant's plate barding. In terms of complexity, real world plate armors consisted of at least twenty individual pieces which were each put on separately, which is why it takes so long to don and doff the armor. We also know that some of these pieces, such as the gauntlets and helmet, can be considered objects in their own right. Thus it is clear that the spell is designed to interact with an item that is both large and composed of many parts, while still treating it as a single object.
Your bard cast heat metal on the Hull of the Illithid skiff; since this is treated as a single object for the purpose of damaging a water vessel, treating it as a single object for heat metal as well seems fair. On the first round of casting, yes, the Illithid crew of the skiff would take damage, whether it was one, two, or more of them. However, on subsequent rounds, if they could move away from the hull or deck, even climbing up on benches perhaps, or moving to the area or item that controlled the skiff's movement, these would likely count as separate objects from what your bard had cast heat metal on. Additionally, given that these are Illithids, one notes that they have the ability to levitate at will - and thus simply hovering over the deck should be enough to protect them from the heat.
Objects are not limited by size.
Objects are vaguely limited by complexity; an object may be made up of other objects, but ceases to be an object once it is made up of 'many other objects'.
It appears that objects are defined by PC's interacting with them and may exist at different, and changing, scales.
A "hull" may be treated as an object for the heat metal spell.