Should an old fat guy with back pain consider a kayak?
Yes!!! Paddling is very low impact. Not like weights. Its as easy or hard as you wanna make it. It may strengthen your back so the pain goes away.
Yes it's relaxing for us old guys with back pain, but I'll tell you from experience you need a kayak with a comfortable well cushioned adjustable seat so you can re-position frequently.
Also for an old guy with back pain you will need to think about the weight of the kayak and what is extremely important is how you will transport it.
Even lifting a 70# kayak one end first into a truck bed can be a chore that can hurt you.
A trailer would be my recommendation if you have any kind of back problems. Then the weight of a kayak would be a minimal issue.
Most people will try to sell you on a certain brand and not consider the above mentioned concerns but it's best to try out as many as possible and find one that you can sit in comfortably for a few hours at a time.