The problem you describe sounds like corrosion on the battery terminals, as well as the inside of the wires running to Engine Ground and off the Positive post. My sister's Previa had the same problem yet if you have a 12 volt booster box, hook it up to either an extra set of jumper cables and then hook the Ground clamp to anywhere on the Engine Block and if possible, turn the key on and touch the Positive clamp to the Starter. If this works, then replace the cable ends or the entire cables, but that can be expensive. NAPA had Marine Terminal ends that worked with the stupid screw down cables attached to the Positive cable end.
If you can afford a friend (IMG:
style_emoticons/default/grin.gif) , have them turn the key...as you stated, but listen to the Battery for a sizzling sound. Even though the cables will look good, the sizzle is high Resistance due to the corrosion inside the cables. Baking Soda & Water work well as long as none gets inside the battery for that will kill it. Also carry a can or bottle of Coke® with you to pour on the cable ends in case of just such an emergency again.
Hope this Helps...
~Stillin~
"Still Living on the Right Side of Dirt..." (IMG:
style_emoticons/default/harhar.gif)