Agreed with what most others have already mentioned;
The Arduino as-is is not suitable for low power operation - it is best to turn it off completely. The circuit below should do what you want, and is based on a very simple and cheap IC, the 555 timer.
simulate this circuit – Schematic created using CircuitLab
A similar version of this has been working for me for about 2 years in a remote sensing setup. A few things to note to make sure that it works;
This needs the TLC555 chip which is a CMOS IC. The TTL version (NE555) will not work with such high value timing resistors.
The timing capacitor (shown as 141uF, just use 3 x 47uF in parallel) needs to be a tantalum capacitor. The normal electrolytic version has too high a leakage current.
As shown, this should turn off the Arduino for about 580s, and on for 120s. To turn off the Arduino sooner, drive digital pin 5 high. This will discharge C1 through R3 and Q1 quickly (about 500ms).
To change the delay times, alter R1, R2, and C1. The datasheet has all the equations that you need for this.
it would probably be wise to put a bypass capacitor between the supply rails very close to the TLC555 chip. 10uF should do.
This circuit should draw around 360uA while the Arduino is off.
One final thing - the task that you are using the Arduino for sounds pretty simple - driving a digital line high for 4 seconds every 10 minutes. I guess that you have more to it than that, but if not, then you could save yourself the cost and complexity, and do away with the Arduino completely. In that case, R1 should be about 6M1, and R2 about 41K, you can remove Q1, D1, R3 and R6, and whatever you are driving goes in place of the Arduino.