interesting town to spend 5 days, no car

Where are you going on the circle tour? How many additional nights will you add? Five or six nights is vague.

The three places that I spent more time in that are easily accessed by train were Killarney, Galway, and Belfast. Killarney has the national park, and many other sights nearby, Galway is the place where you will be able to get to the Aran islands, Cliffs of Moher, Burren and Connemara areas, and I heard some great music in a pub there. Belfast has an interesting history, a castle, city hall is very pretty, and there is the Titanic museum. Plus you can get a day trip to the Giant’s Causeway and other places. Without a car you will have to use guided day trips from to get to sights as public transit won’t work, but all three of these places have many options to do that.

Consider splitting the time (3 days each) between Belfast and Dublin. Both offer a lot, and you can take day trips tours from each to a number of places.

Have you thought about Kilkenny?

A couple ideas:

Dingle Town. I recommend this town because it meets a couple of your criteria: history, good food and walking. The large number of Celtic, Christian and Norman ruins throughout the Dingle Peninsular are out of the world. You could use Dingle Town as a bas. My three favorite ruins are Minard Castle (http://www.megalithicireland.com/Minard%20Castle.html), the Gallarus Oratory (https://en.wikipedia.org/wiki/Gallarus_Oratory) and the Beehive Huts (https://www.atlasobscura.com/places/fahan-beehive-huts). And those are just three of the probably 1,000 antiquities on Dingle. If you won't have a car, you'll have to hire a car or take a bus. There's a hiking trail called the Dingle Way (https://www.dingleway.com/). I've hiked the entire 100-mile-long length of it, but you could use it for a day or two. Dingle Town only has a permanent population of maybe 2,000 people. The negative is that it's packed with tourists in summer, but the popularity of a place shouldn't zero it out either.

Westport: One of Ireland's best-kept secrets. You could climb Croagh Patrick (https://en.wikipedia.org/wiki/Croagh_Patrick), the holy mountain of Ireland. A reasonably fit person could handle it: It took me 2.5 hours to climb. You could also visit Matt Malloy's Pub (of Chieftains fame) and see Rockfleet Castle (https://en.wikipedia.org/wiki/Rockfleet_Castle), which Grace O'Malley, the pirate queen, is said to have successfully defended against a British onslaught. Beautiful, quiet town.

I also second Kilkenny, which was mentioned in another post.

Other ideas: Sligo, Letterkenny and Ennis.

Belfast, Wicklow, Galway

I think Galway is great. Walkable, tourist-friendly, university town. Dingle is just too far from Dublin and it’s pretty sleepy after a few days.