If you expect temperatures ~0c with the possibility of rain you'll need a tent and a decent sleeping bag. If the temperature can be expected to go much lower than that, you need a good sleeping bag. If you don't already have these, then there's a reasonably good chance it'll be cheaper to take a bus.

CM

If you're going off-road (through the woods or something) I would suggest you don't go alone. If you really want to, try to stay in areas with cell-phone coverage and leave note of where you're going and when you expect to be back..

Just get a decent quality sleeping bag. It will keep your body temperature up, and will cost less than $75. I would wear a hoodie with a heavy coat on top, and a beanie to keep my ears/head warm. I assume you'll be going through snow since it's so cold, so some kind of water proof boot would be nice also. If you can remember that the sun rises in the east and sets in the west then you should be able to get to safety with ease if you can remember which direction you were going.

Winter camping can be very dangerous. Get an awesome coat, wear thermal underwear (or two pairs of pants), weat a hat, a hood, a scarf, the works. Carry a backpack with heatpacks, granola, and a big waterbottle. This might seem like a good idea until after the second or third hour =) Long-range hikes are not something for amateurs, however - if you are determined - and it will be more along sidewalks than mountains - I bet it won't kill you to try it.

You can live for a long time on granola, and longer than you think without food. You'll need a constant supply of water though, so if it's sidewalks, refill that bottle in public restrooms or restaurants. If it's offroading, bring iodine-based purifier tabs so you can drink melted snow or riverwater. Don't refill your bottle in standing water (like a pond or big puddle...big puddle *might* be okay if it *just* rained.

If you have no umbrella, strap your hood on tight thru the rain and don't sleep in it. Bring a trashbag to keep rain off you if you have nothing else...water soaks thru the best of jackets but can't do crap to a trashbag. Just secure it over your head and shoulders. You won't look awesome, but this doesn't sound like it's about looking awesome.

Make sure you're comfortable not showering for like a week or two.

If it's in the wildnerness, bring stuff to start fires, and be very careful about doing so. Cheat...use lighter-fluid and carry tinder. Don't sleep next to fires. Careful about sleeping, especially if you might get snowed on; build a little shelter or something out of big sticks and little sticks...roof it with pineneedles.

If it's on the street, sleep under the cameras in ATM booths, and go to sleep around 2 AM...cops likely won't notice you until well into morning (unless it's a major city, then you'll have to fend off bums and, frankly, a motel would be far safer).

I'd make sure you're in shape first, if timing isn't an issue. Start jogging every morning. You might find that the jogging itself helps you feel resolved =)