starter won't spin the engine with all spark plugs

I've got a kit car that uses a Ford 302 V-8, and it has sat for a couple of years and now won't start.  I know the battery is bad, but with jumper cables I turn the key and the engine only turns over about a "little"  (which I define as the fan only turns about a quarter or half a revolution-- about 1 to 2 seconds of cranking and it stops, even though I'm still trying to crank with the key).  I remove all the spark plugs, and try again, and now the engine cranks and seems to spin just fine.... I've cranked it as much as 15-20 seconds continuously.    So then I tried putting half of the spark plugs back in (all of the plugs on the left side), and added a little starting spray in the carb, and low and behold the engine started and ran on just those 4 of 8 plugs.  But I put in the ones on the right side (all 8 installed) and the same problem.  The engine only turns over about a half of a turn on the fan.    Any ideas what could be the problem, or what the next diagnostic step is?