Travel alternatives in Ireland

Hire a private driver for segments or days when you need transportation (that is what we did several years ago, and it worked out beautifully). Then for days when you are staying put, you will not need a driver. In what parts of Ireland will you be traveling?

The Irish rail network connects Dublin to many of the major cities (am thinking Galway, Belfast, Killarney, Limerick, etc.), from which you could settle in for a few days and book day trips to nearby attractions as the spirit moves you.
Once in Galway, for example, you could explore Connemara, the Cliffs and the Burren at your leisure by selecting from a host of day trip providers. Also, the Aran Ferry out to Inishmore has a pickup point right in town which makes the whole excursion about as simple and painless as can be.
Ireland's train system isn't the most robust in Europe but it will get you to (or at least close to) most of the places of interest in the country. Check the map on the website for ideas: http://www.irishrail.ie/. I think you'll find that you can still craft an enjoyable itinerary without a car.
And, as Maggie says, booking a driver for a customized private tour at your own pace (around the Ring of Kerry for example) is also a terrific option.

If public transportation can get you to Kenmare or Killarney, you might want to check out private day tours with Paul Brown for the Ring of Kerry and/or adjacent Beara. His website: http://www.kenmaretaxitours.com/about-us.html

We regretfully had to cancel a ROK day with him in May 2018 due to a medical issue, but he receives superb reviews. If you put his name in the Search field, above, I suspect you'll get detailed info and reviews.

just found out my husband, who will be 80, is not eligible to rent a car

Who told you that?

Enterprise for one has no age limits in the UK or Republic of Ireland. It is fairly rare for age limits at all in the UK (although some load the insurance or require certificates). More common in Ireland, although not with everyone ...

We haven't encountered an age limit in Scotland (or England, for that matter).

Haven't rented in Ireland, but when we looked into it we got the impression that some (most?) companies wanted a letter from your insurance company stating that you haven't had an accident in X amount of time, plus a letter from your medical provider stating that there's no medical reason why you shouldn't be driving.

See other threads on these forums about car rental age in Ireland.

Ireland rental car age requirements are generally more strict than most other countries. The absolute minimum age allowed is 21, but most suppliers require a 23-25 year old minimum, and many will not rent to anyone over 75. If a driver under 25 is permitted, there will be a considerable extra fee (around €30 per day), and drivers over 75 are required to provide proof that they are in good health and have not been involved in any accidents in the past five years.

Thank you for all of your good suggestions! The company we've been dealing with now tells us my husband CAN rent a car, so we're going with that option.

I just want to add, that you might enjoy thinking about not driving in Ireland. I am sure that your husband is a good driver, but many people find driving on the left on narrow roads stressful. The driver can be so busy focusing on driving that he/she misses the sights.

Northern Ireland has great public transport. Republic of Ireland has some as well, but you might need to supplement a little.