If the engine will not crank over, the most common problem is the starter. These will tend to go out at about 100,00-125,000 miles. Sometimes it is only the starter solenoid, but often the complete starter (including solenoid) needs to be replaced.Google+
Car Problem Reports
Toyota Avalon Vehicle Won't Crank Due To Starter
Toyota Avalon Problem
Average Mileage: 127,500 mi (110,000 mi - 150,000 mi)
Visitor, 1995 Toyota Avalon, 130,000 mi
Car wouldn't crank, had to replace starter. Only took me an hour or so to pull the old and put in a new one.
Visitor, 1995 Toyota Avalon, 120,000 mi
Had to replace starter. But it is so easy on this model. You don't have to crawl under car. Just remove the air box for easy access.
Visitor, 1997 Toyota Avalon, 150,000 mi
new starter, new alternator, and a new battery