How do people manage to find places to live where $400 a week is enough for rent alone, let alone groceries? Or do I just live in the worst possible area, paying $1600/mo for a small apartment, then having to pay for utilities too, and another $100/mo for internet, and that's not even accounting for groceries.

However, $400 a week for a month would be almost 3x the costs of the above, so some of that would go to rent.

So if your monthly spend is $2,500 and you get $1,600 from unemployment, then you only need to spend $800 of your savings each month, not $2,500?

You just extended your runway by 3x.