The Homeless Survival Guide, page 3

This is a thought that's crossed my mind more than once. Aside from the fact that it really can happen to anyone in almost no time flat (in the worst case scenario, almost nobody is more than 2 months from being homeless- for many of us it's more like 2 weeks), there are non-financial reasons a person might be forced to forego keeping a permanent address.

If I were going to become homeless, or just had become homeless, my first move would be to put a call into the financial aid office at a community college to see if federal, state, and other assorted aid could put enrollment within my means. It's not an option for everyone, but for some it is. Obviously very few people think, "hey, I'm about to be homeless, what better time to apply to a college" but in some cases, it does make sense.

My experience is only with California so I don't know what it costs elsewhere, but around here, to carry 12 credits will cost you about 500 your first semester and 350 each subsequent semester.
So about 100/month- which is irrelevant because if you're going it all you're going on the state's dime, at least in large part. If you get out ahead of your upcoming homelessness you could theoretically even do it in a ridiculously tiny student loan.

For that, you get:

Shelter during the daytime- some places it's OK to nap, some it's not. Student centers are usually good, the library depends on the staff, and classrooms that are empty for a while before your class starts are good sometimes too. Depending on weather and insect life, there are usually shaded lawns where you can lay around unharrassed as long as you're looking student-ish too. You see a lot of sleeping going on around college campuses. It's amazing anything else ever gets done.
Basic hygiene facilities in the locker room- which is also a place to story a few things so you aren't lugging a huge load during the day.
A shot at relatively well paid, though very part-time jobs that are reserved for students.
Access to a career center.
A chance to do some networking that could eventually land you a job, a place to stay, etc.
Occasional events featuring good free food- watch the clubs like a hawk.
Computer access, which lets you stay in communication for free, and also helps the job hunt.
The right to apply for scholarships, etc that you can spend on living expenses.
A new photo ID (helpful if your state-issued one is expired).
If you're desperate, you might get away with scavenging trays that were at cafeteria tables, although that's conspicuous and would be a bad habbit to get into.
If you actually end up stuck there for the duration- job retraining if there's an in-demand field you need help getting into.

Not having an address is less of a problem than you'd think for that. In my experience nothing they send you HAS to be received. You could give an old personal address as your permanent, but claim a rural address as your current and have them send you your mail general delivery- and you can pick it up or not as you see fit. In fact depending on what you get into at school, you could have mail that you MUST have delivered to you care of a club on campus that you join, if you check that out with the honchos first.

So your remaining needs are basically shelter at night and food, soap, laundry, and if you're male, cheap razors.

A food budget can be squeezed down pretty low- somewhere in the vicinity of 60 cents a meal if you can acquire a taste for cold Ramen noodles and assorted canned stews, beans etc. Razors you might do as well as .75 to a buck each, and get 3 uses out of each one, shaving every other day, and hopefully you'll have enough in your wardrobe to get through a week (5 shirts, 5 socks/underwear, 3 pants would do it) so you're not doing laundry TOO often so hygiene ends up looking like 20-25 a month if you're using a coinop laundry, which I actually would recommend, and food somewhere in the neighborhood of 60 a month.
Making $85/month is not exactly a daunting task (you could literally make that just selling blood plasma, though recycling etc seem a lot more realistic).

Shelter at night can be tricky if you mean to stay away from homeless shelters, which would definitely be my preference in such circumstances. If you own a vehicle when you become homeless, you are allowed to spend the night in Walmart parking lots. You'll have to move it around the centers parking lots daily so it isn't towed as an abandoned vehicle, but that method will last you a few months anyway, until it either breaks down or eventually the tags expire and it gets towed. Don't keep anything you want in there.

If you don't have a car, which would seem more likely, I'd personally plan on sleeping in the afternoon and early evening and being up nights and mornings, assuming climate made it feasible to be out at night. That allows you to avail yourself of acceptable public areas for napping. Of course that presents the problem of what to do when you're awake at night, but it's not a perfect world. My next choice would be a tent outside city limits, again climate permitting.