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.
Car Problem Reports
Toyota Avalon Vehicle Won't Crank Due To Starter
Toyota Avalon Problem
Average Mileage: 140,000 mi (130,000 mi - 150,000 mi)