Battery is 12.6 volts minimum standing voltage. If you have less than that and it is a true indicator of the state of charge of battery, then you are on the slippery side of the slope.
Taken from the battery university web page, this heads the nail on the head:
specific gravity Open circuit voltage
100%1.265 2.10 6.32 8.43 12.65
75% 1.225 2.08 6.22 8.30 12.45
50% 1.190 2.04 6.12 8.16 12.24
25% 1.155 2.01 6.03 8.04 12.06
0% 1.120 1.98 5.95 7.72 11.89"
These are measured after 24 hours of rest, to let the battery normalize, but this can be an indicator of internal battery issues. You short one plate with another plate and you lose, 2.1V+ at 2 cells, being 4.2V of the 12.65V, leaving you off the chart and unable to start. Take the battery in to be tested on a capacitance tester - all the major chains should do this for free I suspect.
Other than the battery, you would be likely in the area of battery terminal cleanliness and as Thom said, checking your grounds. The battery cables should be checked for resistance too, as it would only get worse with a load on it.
If you connect a DVOM in series with your positive wire and the battery and hook the battery back into the equation (but do not have the key place at all!), see what kind of amp draw you get. Be sure to have a 10A model and keep a spare fuse around just in case you pop it! Be on the 10A setting not 1A and see what you get. If the draw is so small it doesn't accurately show on the 10A scale, then move to the 1A to get multiple decimal places for output.
We call this parasitic draw and anything from a failed diode in the alternator rectifier bridge, to a dome (those pesky door switches and sagging door hinges! or trunk or glove box light still on, to a module or ECU could be pulling in excess of what would be acceptable draw for most cars.
I vote 20-35 mA tops, anything more and you are looking at trouble or you own a corvette or a german vehicle that normally has a rather large battery anyways with a great RC rating.