Shot starter?

Make sure when you hit the key you have at minimum 10.5 volts on the big starter terminal if it drops away to zero then comes back with the key off you have either an open cell in your battery or poor connection (usually ground). Also you can check starter draw and load test battery I just changed out my battery for similar symptoms. (My battery was good and started my car at 3:30 and at five it was junk)