I am highly against buying any insurance outside of your homeowners or rental insurance. Homeowners or rental will cover you in the case of a disaster or other event that damages your appliances.
The third-party policies to cover appliances are typically highly expensive and rarely worth it. Considering your appliances are already 20 years old, it is highly unlikely that a policy would be allowed unless you paid a higher premium...definitely not worth it.
What I would suggest is taking a fixed amount that you would feel comfortable paying to an insurance company, and putting it in a savings account that you ABSOLUTELY do not touch unless one of the appliances dies. The money stays with you the entire time and you don't risk losing it to an insurance company.
My wife and I did something similar years ago in an apartment we lived in. The unit came with a washer/dryer and we were used to going to laundrymat. I figured we spent anywhere from $10-20/week on laundry so each week I put aside $20 in a savings account (very thankful for automatic withdrawal). After over 2 years, the dryer died and the landlord did not want to replace it. We had almost $2000 in savings and I told him we'd buy the washer and dryer if we could use their hookups. He agreed and we still have those applances today! You can do that with just $20/week!