The 4.4 kg I gave was carbon, the 16.3 kg CO2. Your 9.16 kg refers to CO2, fixed per square meter per year, which is less than the CO2 fixation value I'd used. That is, http://carbon.ycombinator.com are making a more conservative estimate of about 56% the maximum observed rate. That increases rather than decreases the required area.
At 9.63 kg/(m^2yr) you'd need 39.8 m^2, or 429 ft^2, of algae to offset one person's CO2 exhalation.
Using GNU units:
You have: (1kg/day)/(9.16 kg/(m^2*yr))
You want: m^2
* 39.873602
/ 0.025079249
You have: (1kg/day)/(9.16 kg/(m^2*yr))
You want: ft^2
* 429.19589
/ 0.0023299385
You have: 9.16/16.3
You want: %
* 56.196319
/ 0.01779476
> We will be conservative and say that our algal beds fix 2.5 kg of C per square meter per year and 2.5 kg of C works out to 9.167 kg of CO2.
But then a couple of paragraphs later:
> the maximum CO2 assimilation rate of algal beds which we will say is 9.167 kg of C per square meter per year.
I.e. they first mention a conservative "2.5 kg of C" in the context of the ocean, followed by a "maximum rate". 9 kg of carbon is around 33 kg of CO^2.
Maybe they meant "kg of CO2" in that second paragraph, it's certainly suspicious that it repeats the same 9.167 number, but why would they call that number both "conservative" and "maximum assimilation rate", hrm...
In terms of CO^2 sequestration it doesn't seem implausible to fit a farm of that area in something the size of an apartment..
Of course the energy/water use would be insane, hopefully you could funnel sunlight in, or just open the window. But as a thought experiment that ignores the economics of it think more space station than apartment.
Critically, you need on the order of N m^2 of sunlit area on your culture, not N m^2 of floor space. If you image search "phytoplankton culture" you can see that people grow these in bottles or thin plastic bags under artificial lighting, maximizing the surface area.
You can easily fit 11 m^2 in one square meter like that under standard 250 cm ceiling height. Just have a stack every 20 cm.