1997 Corolla would turn over

I have a a '97 Corolla with 190,000 plus miles on it. Today the motor would not turn over. Also had a slight smell of fumes but no leak. Any ideas of the cause and the fix?
my '03 with 155,000 has similar problem, from only a few days ago. battery tested, and it is good.

latest analysis: starter relay not working. Problem now, just HOW does one get it out? wiggling does not
work, altho' it did with the adjacent relay (for turn signals)

IF anyone out there can advise, you may be helping TWO people...