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I normally ask sci-fi questions, but oh well...

It's fairly well established by this point that gills, while fine for fish, wouldn't work for mammals, as we need more oxygen than can reasonably be found in the water. This goes double for humans, since our brains use a lot of oxygen. Whales, however, don't use gills. Whales breathe on the surface, and stay underwater for anywhere between 20 to 90 minutes. And if you take a look at the tail of a traditional mermaid, you'll notice that it actually resembles the tail of a whale more than it does a fish (the exception is shown in the fourth Harry Potter movie. Those merfolk had fish-like bodies, which made them a lot creepier). If we go with the traditional depiction of a mermaid/merman having a human upper body and a whale-like lower body, it strikes me as being more realistic that merfolk would be air-breathers that need to come up periodically for oxygen.

We don't have to explain how merfolk could have possibly evolved. Let's assume that they were created via magic, or genetic engineering. Let's also assume that that upper body only has to LOOK human, internal organs can be rearranged. For instance, the upper body could be filled by a massive set of lungs twice the size of a human's, while the digestive tract has been mostly moved down into the whale body.

With this in mind, is it physically viable to have air-breathing merfolk who can routinely stay underwater (actively moving around, not just sitting to conserve oxygen like human record-breakers do) for at least 30 minutes, or are there too many biological problems with the whole concept?

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Your 30 minutes of underwater activity limit is tough, but it is eminently achievable... as you've noted, whales can already manage dives far in excess of that. Even looking at something a bit more human in scale, such as the habour seal, you can see that they're capable of diving for 30 minutes.

For instance, the upper body could be filled by a massive set of lungs twice the size of a human's

Seals and their relatives exhale before diving. They store their oxygen in myoglobin (closely related to haemoglobin, but specialised for storage rather than transport) which is found in much higher levels in their muscles than in yours. There are other adaptations in other species which could be handwaved in, such as crocodile haemoglobin which is better at releasing oxygen than yours is.

You haven't specified diving depth, I note, and that is important. Seals and walruses and the like tend to dive deep, and for them having full airspaces that could get squished at depth is undesirable. For a shallower-living species, the ability to keep their lungs inflated gives them a little extra oxygen and a little more CO2 rejection capacity and both these things might help extend dive times.

are there too many biological problems with the whole concept?

The major problem is habitat. Water sucks heat away quickly, and if your merfolk don't live in the warmest, balmiest places they can find, they might have problems. That harbour seal I linked above? Pretty round and a bit furry, you may note. Lots of blubber. Any mermaids (or merblokes, for that matter) you might meet in the north Atlantic, sitting on a rock and brushing their hair and singing away will have a generous layer of what some divers term "bioprene". "Rubenesque" ain't gonna cover it.

Those arms are also quite unhydrodynamic. Or that hair, for that matter. For a given time period, fairly human-looking merfolk aren't going to be able to swim as far or dive as deep as better adapted marine mammals, though they could use tools and have a slightly easier time of getting around out of the water.

The basic idea doesn't seem impossibly far fetched, though.

FWIW, I always assumed that merfolk were not fish... the females are famously mammalian, after all. Sure, they're traditionally drawn with a scaled tail, but so were whales and even giant cephalopods, back in the day. The illustrators of antiquity were not necessarily very rigorous.

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    $\begingroup$ I figured that the heat problem would be an issue... it didn't occur to me that whales exhaled before they dived, though it really should have, considering that I know that sperm whales routinely dive down deep enough that they'd be under several dozen atmospheres of pressure. So merfolk would probably have human-sized lungs, and spend several minutes on the surface between dives letting their myoglobin reserves replenish. Realistically, their rib cages would probably need to compress in order to let the air out for deep dives, and looking at THAT would probably be quite creepy. $\endgroup$
    – FlyingLemmingSoup
    Jan 15, 2020 at 20:17
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    $\begingroup$ @FlyingLemmingSoup not all cephalopods exhale (dolphins don't, not sure about the rest) but seals generally do. Ribcages don't collapse, but there are all sorts of other weird physiological changes (read up on bloodshift in freedivers, for example). Windpipes on the other hand do fold flat in some species, because you don't want rigid windpipes being crushed at depth... $\endgroup$
    – Starfish Prime
    Jan 15, 2020 at 20:20
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    $\begingroup$ It occurs to me that if merfolk can't swim as fast as other sea creatures... that actually makes them the underwater equivalent of, well, humans. Humans aren't particularly good at any individual survival trait, apart from intelligence, tool use, and endurance travel. We're slow runners, slow swimmers, slow climbers, weak, no natural weapons... merfolk would be the same way, compared to the critters in their environment. They're slower than fish, and have no natural weapons the way a sharks do. The difference is that they're smart, and can create tools to make up the difference, like us. $\endgroup$
    – FlyingLemmingSoup
    Jan 15, 2020 at 21:28
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    $\begingroup$ @FlyingLemmingSoup note that humans are slow runners, but with high endurance and excellent heat rejection. They're slow swimmers, but good enough to be able to hunt and gather in an aquatic environment. Poor climbers, but they can thrive in places where there are no other climbers. Human adaptability is the key trait, there. $\endgroup$
    – Starfish Prime
    Jan 16, 2020 at 8:22
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    $\begingroup$ What makes humans such good persistence predators (among other things) is that we can run very efficiently compared to other animals. In order for your merpeople to fill the same role they'd need to be similarly efficient (which means the hair and forelimbs are not really desirable). If they're able to fold their forelimbs flush with their body, and lose the head of hair then that would be much better. $\endgroup$
    – Ynneadwraith
    Jan 17, 2020 at 13:13

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