When there is a heavy rain water backups into the home through all the drains. What is the likely cause and is there an inexpensive fix I can try first? 40s era single story no basement.

Sounds like you have a partial blockage downstream from the connection of the sewer and the rain water run off, The solution would be to snake from the clean out to the street,
Another solution would be to install a sewer check valve ( see image) just after the house cleanout. And a third solution would be to install a Flood Guard (see image) on your floor drain. Check them out at Flood Guards For Floor Drains - check valves to prevent flooding
Our code outlaws anything discharging into the sewer but sewage and gray water, Good luck, Tom