Resistance, and learning for beginners
The wire suggested wire is adequate- the resistance is about 130 ohms/km. I got that from kinda a long route. I used the table for copper (AWG) wire resistance per km, adjusted that for the ratio between copper and steel resistivity, and then for the ratio (squared) between the diameters for AWG and the gauge system used for steel wires (whew!) which is "Washburn &amp; Moen; Roebling; or American Steel and Wire".  

So only maybe 2.5 ohms for your circuit (based on two wires, 9.1m each, connected at one end. 

You might want to use something other than wood for the supports, insulate the wire from the posts, or at least seal the wood with epoxy. If, say, you were in Portland OR, it will (on average) rain 144 days per year and the leakage through the wet wood would tend to drain the battery. 

Here are some insulators used on commercial electric fences. 



In terms of $ per kWh, AA batteries are a much better deal than 9V batteries, and six AA cells in series would give you 9V. Energy capacity being a lot higher, it also means that you won't need to replace the batteries as often. There is a good reason why 9V batteries are seldom found in modern electronics. 

AA cells have a short-circuit current in the amperes, so a current limiting resistor as Andy suggests would be a very good idea, or you could use a blinking LED in series such as this Lumex one, available from distributors such as Digikey for $1 each. 



If you see the LED blinking, the wire is shorted.