Sears Kenmore 70 series electric dryer - runs for 5 minutes and then shuts off - any ideas why?

My guess is that it is getting too hot and shutting off for safety.

I would pull it out from the wall and clean the exhaust hose (and everything else you can reach) of lint. Also, clean out the hose that goes inside the wall -- all the way to the outside.

That will most likely fix the problem. If not, then you may have a worn-out thermal limit switch, or some other component like that.

Hope this helps,
WallyH