My kenmore 70 series washer filled up with water and then stopped

This assumes that you have a basic Kenmore / Whirlpool top load, direct drive, neutral drain (it doesn't spin until the pump removes most of the water) machine.

First, see the Sears parts site for your washer:
http://www.searspartsdirect.com/partsdirect/index.action

Enter your model number (###.######) and you'll see a list of major sub-components with diagrams and parts.

If the washer does nothing after filling, I'd try it on a different cycle.

If it works on another cycle, the problem points to the timer itself.

If you can hear the motor trying to run but the washer will neither agitate nor spin, the problem points initially to a Motor Coupler.

Kenmore / Whirlpool top load, direct drive washers use a "Motor Coupler" between the motor and transmission. It's used as a fail safe device and consists of three parts. Two of the parts are the same, plastic units with three prongs that fit on one of the motor output shafts and the other on the transmission input shaft. Between them is a rubber piece with six holes. The power from the motor is transmitted through the rubber piece to the transmission.

The other shaft on the motor directly drives the pump. This is likely why the machine drains.

If the motor coupler is broken, or worn, the motor may not be able to send full power to the transmission for agitating or spinning.

See the following for how to remove the cabinet, pump, motor and motor mounting plate to inspect and replace the motor coupler.

http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=137

If the coupler looks damaged, it's a relatively easy fix.

Pry the plastic pieces from the motor and transmission. Some people use a claw hammer but I've found that using a small (6 inch) pry bar works well.

Carefully fit the plastic pieces over the motor and transmission shafts and press them in place (Use a 1/2" of larger socket to put pressure equally around the pieces.

Put the rubber piece on the transmission plastic piece.

Replace the motor mounting plate.

Carefully line up the plastic prongs on the motor based plastic piece to the three un-used holes in the rubber piece. If they don't line up right, the coupler could be damaged.

Mount the motor, don't forget the two screws, put the pump back on the motor and use its' clips.

Put the cabinet back on, don't forget to plug the lid switch back in and re-mount the console.

Before tearing it down and replacing expensive parts you might want to make sure the door activated switch is not coming apart. They are known for this. I have had to fix 3 already.