Battery life with Nano v3
Something is wrong with your setup. I don't have a Nano to hand to test, but it would consume similar current to a Uno (probably less) which is around 50 mA. Thus, using a 2800 mAh battery you should have a life of 2800/50 = 56 hours.

Li-ion batteries come in multiples of cells, where individual cells have a nominal voltage of 3.6V (thus you probably have three cells giving 10.8V). Deeply discharged they will have a voltage of 3.0V per cell (a total of 9V for your three cells). If you started at 9.8V as you stated, then the cells would have been 90% discharged already which probably accounts for the short running time.

Assuming you haven't wired it in a strange way (and it would help to show the wiring and possibly a photo) I suggest you try with a fully-charged battery.



Another helpful thing to do would be to measure the current consumption of the battery using a multimeter. Make sure you read up on how to measure current with a meter or you may damage something (example here).

The basic idea is shown in this photo:



The meter is placed in series with the battery and the Arduino. The negative side of the battery goes to the Arduino in the usual way. The positive side of the battery goes into the "milliamps" input of the meter (which is set to 200 mA range in my case) and then the common (black) lead from the voltmeter goes to the positive (Vin) side of the Arduino.

You can see that in this particular case, with a 9V battery, the Arduino is consuming 57.7 mA.  I don't quite believe that reading as re-measuring with a more expensive meter gave a reading of 51.1 mA. Still, that gives you a ball-park figure. And it shows that spending a bit more on a better meter gives you more accurate results. :)



I have a page about power saving and you can also refer to the Wikipedia page about Li-ion batteries.




  my volt meter read 9.8v and 1 hour later 2.8v. causing the Nano to go off.


For a 12V Li-ion battery get down to 2.8V there is something seriously wrong with its protection circuitry. It should cut the battery out at 3V per cell, that is at 9V if you have 3 cells in series.