My apartment receives hot water from a shared hot-water heater, and taking a hot shower warms up the 100 m³ apartment noticeably. So I filled up a large plastic tub and dragged it into the living room.
This plastic tub is 330 × 300 × 600 mm, more or less, and it’s full of 40° water while it’s 15° outside. This is about 60 kg of water, which has the thermal mass of about 120 kg of air, and about 1500 kilocalories or 6 megajoules. If this tub takes 6 hours to cool off, which seems plausible, it’s heating the apartment at almost 300 watts on average during that time.
To heat the apartment with the hot water supply at the 2000 W or so it really needs, I would need about 19 milliliters per second of water, or about 70 liters per hour, considerably below the shower’s output of some 300 mℓ/s. The hot water flow would need to be intermittent, because the first liter that comes out of the faucet is cold, and the first ten or twenty liters are noticeably below maximum temperature — drawing water continuously at 19mℓ/s would just warm up the pipes in the wall, not the interior of the apartment.
Refilling one of two 35-ℓ hot water tanks every half hour would take about 2’ out of every 30’, with the final result being equivalent to a 90-minute shower every day.
So my shower is heating my house at 300 mℓ/s · 25° · 1 kcal/kg/° · 1 g/cc = 31 kW. That’s pretty respectable, especially given that the electrical service is only 66 amps, 16 kW. Too bad that, in its current form, the shower also raises the humidity uncomfortably high.
The hot-water tanks could take the form of tall cylinders with hot water at the top, cold water at the bottom, and a heat exchanger in between, driven either by thermosiphon action or by an actual pump. At 140 mm diameter and 2.5 m tall, they would hold 38.5 liters and thus 4 MJ of heat at ΔT = 25°. By separating the cold water from the hot water in the same tank, they would avoid the need to have double the tank capacity to separate hot from cold, or to put air into the tanks between emptying and refilling, which can promote rusting.
However, if you just left them uninsulated and didn’t use a heat exchanger, the result could be adequate with a bit more work. Each tank as described would have a surface area of 1.1 m², and at 40°, a black body radiates 545 W/m², so this would be about 600 W of heat emission per tank — counterbalanced by absorbing 430 W from the environment, so only 170 W net. This is about an order of magnitude too low, suggesting that you could get the right result by flattening the tank out to be an order of magnitude thinner and thus wider.