The tankless water heater can be mounted with either concentric venting (preferred) or dual-pipe venting - one for intake and one for exhaust. Because of this, we would not be able to 'common vent' in with the furnace exhaust or even the chimney itself because the concentric would be pulling air down the chimney or suck in the exhaust of the furnace. We could take the exhaust pipe of the dual pipe option and add it to the chimney or furnace, if sized correctly, but regardless we'd still have to direct vent outside for the intake. So, because of this, it makes the most sense to go with the concentric venting and route it outside.
The best location that we're working with is in that old coal storage room. This provides access to 3/4" water lines and the 1" gas line that we can pull off of for the 3/4" line needed for the water heater. The only workaround now is with venting. The gas meter in my picture is actually farther down the exterior wall than is shown (the gas line does a 90 after coming through the wall) and up to where it's shown in the picture. This allows the easiest access to venting outside through this area, although we're required to keep a 3' distance from the gas meter for it's regulator. This shouldn't be a problem because we actually have to run it up the wall about 3' anyway to clear the snow level. If this isn't workable, the other option is to run the vent to the complete other side of the basement and come out of the wall there. This could come close to the maximum allowable vent span, but it's an option currently.
If you're only worried about the water heater, then you can turn it off and drain it.
Shut off the gas
You should see a gas pipe connected to the unit. It will likely either be rigid black pipe, or flexible yellow pipe. Follow that pipe until you find the shutoff valve. It should be a quarter-turn ball valve, so you'll only have to turn it a quarter turn to shut it off.
An easy way to tell if the valve is open or closed, is to look at the handle. If the handle is in line with the pipe (parallel), then the valve is open. If the handle is across the pipe (perpendicular), then the valve is closed.
Shut off the water
Since you're only concerned about the tank, you only have to turn off the water feeding the tank. Locate the cold water supply pipe (possibly marked with a blue ring on the tank). Follow the pipe until you find a valve. In this case, it could be a ball valve or a gate valve. As with the gas valve, a ball valve will only require a quarter turn. A gate valve, however, will require multiple turns to shut.
Drain the tank
Connect one end of a garden hose to the spigot at the bottom of the tank, and run the other end to a drain that's lower than the spigot. Open the hot water valve on a few faucets in the house, to allow air into the plumbing. Open the valve on the spigot at the bottom of the tank.
CAUTION The water draining from the tank will be HOT.
If you have time, turn the tank off and allow the water to cool before draining the tank. This could take a bit, as the tank is designed to hold the heat.
When you get home
When you come back, make sure the spigot at the bottom of the tank is closed. Open the valve on the cold water supply line, and allow the tank to fill. You'll know it's full, when water starts coming out of the faucets you opened throughout the house.
NOTE: As the plumbing fills, air will come out of the open faucets. As the system nears full, water and air will spit and sputter from the faucets.
Once full, close the faucets. Turn on the gas, and follow the instructions on the tank to turn it on (light pilot, etc.).
Best Answer
40°F is above freezing, so freezing shouldn't be a problem.
If the weather is getting cold and you're concerned that this utility room might dip below the freezing point, prop the door open a bit to allow some heated air in.
Also, since this is an apartment, talk to your landlord about your freezing concerns and ask him to address the issue - maybe adding some insulation to all the pipes in this room, or, even better, adding heat to the room!
Additionally, based on reading the comments on the OP, talk to the landlord about your freezing concerns and ask him if this is something for you to worry about at all. It may be that she's fully aware of the temperature issues and purchased heaters that have an anti-freeze feature built right in so this won't be a concern at all. It's still possible for piping to freeze, but again, 40°F is above the freezing point and it's unlikely to be a concern unless a serious cold spell hits and gets this unheated room really cold. Simply opening the interior door to this room should allow more than enough heat from the rest of the apartment into the room to keep it above freezing.