You can then change it to Mariculture, not Aquaculture. Talking about the Current World, this is one of the most promising methods of food production that might improve our condition in food security and to an extent, solving the climate issue (which cannot overlook other solutions like Nuclear Energy and Space, but can certainly be supplemented by Mariculture. Let's discuss this later).
However, this would likely be limited to the coasts, but likely enough to sustain a city state or a small state.
The sort of mariculture you're discussing involves scenarios like massive ocean thermal energy convertors which artificially do the upwelling job ocean currents do in certain areas. And on a smaller scale, very precisely built artificial reefs for seaweed, shellfish, and fish farming.
While it is true that artificial shellfish beds existed among Neolithic-level cultures like the Coast Salish, they involved a lot of investment in labour and maintenance relative to their yield and would be suboptimal by today's standards. Seaweed/other sea plants and shellfish are not sufficient for supporting a large number of people because of relatively low calories and especially lack of carbs. Carbohydrates and too much protein is actually a major limit in these sedentary hunter-gatherer societies. The wild plants and nuts they gather, even those whose growth they maintain, are insufficient for the population size so they rely heavily on fats and oils to supplement it but that only works to a certain extent. This was the case among the indigenous people of the Pacific Northwest, it happened in late Jomon period Japan, and it presumably happened in pre-agricultural Europe.
The problem with making a city-state out of a system like this is that resources are too easy to collect and certain tribes or lineages will lay claim to certain beaches, fishing sites, gathering sites, etc. The institution of a state seems to rely heavily on stored surplus, and in a system like this, every village can store its own surplus so you end up with a large village of a few hundred people with some authority over a dozen or so smaller villages of 50-200 people and usually all within the same drainage basin (i.e. where the salmon runs are and weirs are maintained). This was the political structure in the Pacific Northwest, precolonial southern Florida, and among the 17th century Ainu. Your chiefdom probably does not have more than 8-9K people (although the chief will be extremely prestigious regionally), and the political structure is unstable enough that villages can easily switch loyalties to a nearby tribe on the basis of family ties or an unjust ruler.
That said, I could imagine that
theoretically if you had a culture perfectly suited for it, you could have a city-state of maybe 1-2K people survive a few generations before exhausting local resources and being hit with repeated epidemics because of the sanitation issue. It would probably rule a conventional chiefdom like I described above (but probably with smaller subsidiary villages because of the food issue) and maybe stretch into nearby chiefdoms and command perhaps 10-15K people.
If I had to pick an area, then Southern Norway would be the best for this since it's hemmed in by mountains (which as seen in the Pacific Northwest and among the Ainu is a good factor for encouraging complex chiefdoms) and isn't too far from the Baltic and mouth of the Elbe. They could be indirectly trading with people as far south as the Alps and Carpathians through visiting the deltas of the Elbe and Vistula. This would give the ruling clans of this city some serious status which they could use to attract other local elites to settle in their city, bringing with them their followers. This is the sort of "ritual complex" state which admittedly is usually associated with farming societies (i.e. some Puebloan city-states, or Cahokia) but I don't think it's totally impossible a hunter gatherer society could build such a state. Probably all it takes is the right cultural conditions, a string of lucky marriages and a wise chief born from it, a bit of vision (i.e. erecting some monument that becomes a cultic site), and plenty of luck.
I'd also love to see a Doggerland-based culture doing this in the 7th millennium BC period with the ruling chief's authority based on maintaining primitive dikes to keep the water levies low. Doggerland has the environment to sustain a sedentary hunter-gatherer culture, so it would work too. Obviously it's still going to be utterly screwed when the island is flooded, but they could probably be a successful and wealthy maritime culture until that point. Theoretically if they were into building large earthworks like the Indians of the American South, they could build a structure that would extend above the sea even after it drowns. The highest points of Doggerland are 15 meters below sea level--the largest mound at Poverty Point is 22 meters (and was built by hunter gatherers unlike later mounds). Most likely these mounds (I imagine maybe 3-4 would remain above sea level) would gradually be eroded and vanish over the centuries (as many actual islands and entire cities did), but perhaps remnants of Doggerland's elite flees to England or the continent and occasionally revisit their homeland, and even when the Indo-Europeans overwhelm them, it's maintained as some sort of ancient tradition. Maybe these Celts/whoever put enough maintenance into the remnants that they survive into the modern day (maybe even Christianised as shrines to some saint associated with fishermen).
In both cases, the challenge is that such a city would inevitably depopulate the surrounding countryside to use as a reserve for wood (fuel) and game and the issue of sanitation. Both factors limit the size such a state can expand to and limit its population, but it would still be a very impressive city and one which would undoubtedly embed itself into local myth and religion long after it was abandoned. I said 1-2K people, and that still makes for an impressive size if you consider the level of technology of those who built it and what they might leave behind.