(See also An electric furnace the size of a sake cup for more thoughts along these lines.)
Suppose I want a ceramics-firing kiln that holds about a liter, about a thousandth the capacity of the big kiln in the studio; call it a “millikiln”. Maybe I can make it by packing calcium oxide into a ceramic jar, putting some kind of heating element inside, and cooling the jar on the outside so the ceramic doesn’t melt.
I’m going to guess that 30 mm of calcium oxide is adequate insulation. The inside volume of 1 liter could be 100 mm vertically and 112 mm across; the outside volume, including the lid, could be 160 mm vertically and 172 mm across, a total of about 3.7 liters, which means 2.7 liters of calcium oxide. The most practical way to obtain that is to calcine about 2.7 liters of calcium carbonate, whose specific gravity in the calcite form is 2.71, so that's about 7.3 kg of calcium carbonate.
Is that right? Calcium carbonate is CaCO₃, while calcium oxide (CaO) has a density of 3.34 g/cc, which is higher. And, indeed, WP says, “The reaction of quicklime with water is associated with an increase in volume by a factor of at least 2.5,” while “Approximately 1.8 t of limestone is required per 1.0 t of quicklime.”
This suggests that calcining the calcium carbonate will reduce its volume by a factor of 2.2 to more than 2.5, which probably will not leave it solid.
Also, what about harmful gas emissions during firing? The 2.7 liters of CaO will weigh 9 kg and be made from a bit over 16 kg of calcium carbonate; 44% of the mass of the calcium carbonate (the other 7 kg) is lost as carbon dioxide; 7 kg of carbon dioxide is about 3.5 cubic meters ((12+16+16)/(16+17) = 1.33 times the density of air, which is 1.2 g/ℓ at STP, so CO₂ is 1.6 g/ℓ, XXX wrong 1.98). CO₂ is safe and beneficial to plants at 1000 ppm, but becomes problematic to humans around 5000 ppm (though submarines commonly have it up to 20000 ppm for long periods of time). 7 kg gives you 5000 ppm by mass when mixed into 1400 kg of air, occupying about 1200 m³, which as a sphere would be a bit over 13 meters in diameter.
This is larger than the pottery studio, which is perhaps 6 m × 6 m × 12 m, a total of 432 m³ or 518 kg of air; so it should be safe to release about 2 kg of CO₂ into it at a time, obtained by calcining some 4.5 kg of calcium carbonate.
Perhaps a different refractory would be more suitable. Maybe ordinary fireclay, which seems to cost AR$10 to AR$20 per kg on Mercadolibre, although I have my doubts about whether the various products sold there as “arcilla refractaria” are in fact any kind of clay at all. Maybe perlite, although supposedly perlite is only good up to 1000°, and vermiculite is hydrophilic (though much less so than quicklime!). Some sources suggest mixing in sawdust and burning it out to increase porosity of the refractory without perlite.
(Vermiculite also melts at 1100°C or below.)