Question: Given an Earth-like planet, what sort of geological features or weather systems are required for a place to have the same average day and night temperatures throughout the whole year? Is such a thing even possible?

The closest I could think of were tropical climates that are hot all year long (though even these have cooler wet seasons), but I'm looking for something more temperate, around 15-25°C during the day.

The minimum area of the region should be roughly as big as Belgium.

Answer: 
High altitude tropics
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While San Diego has remarkably stable temperatures year round, there are other places that are even more stable. In tropical highlands, places get the same stable insolation as the steamy lowland tropics, but the air temperatures are mitigaged by altitude. This results in basically the most pleasant places on earth.

There are several tropical highlands straddling the equator. Africa has the Rift Valley mountains, from [Ethiopia](https://en.wikipedia.org/wiki/Addis_Ababa#Climate) and [Nairobi](https://en.wikipedia.org/wiki/Nairobi#Climate) in the Kenyan Highlands through [Rwanda](https://en.wikipedia.org/wiki/Kigali#Climate) and Burundi. All the cities in the Andes from [Caracas](https://en.wikipedia.org/wiki/Caracas#Climate) to [La Paz](https://en.wikipedia.org/wiki/La_Paz#Climate) have pretty stable temperatures. And through extraordinarily wet, the [highlands](https://en.wikipedia.org/wiki/Mount_Hagen#Climate) of New Guinea have very stable temperatures too.
It's called a cave. Go down about a hundred feet and the weather pattern is very stable. It's a bit chilly, but not impossible to make a comfy place.

Avernum takes place in a giant interconnected cave system.

The same could be said for the Underdark, though I've only been there in Neverwinter Nights so I can't say if there are storms.

Journey to the Center of the World also featured interconnected cave systems.

If you want something a little more open, a Dyson Sphere or Ring should also have the same temperament everywhere unless something's gone terribly wrong. Like the heat source drifting closer to one wall (floor?) than another.

Planets can't really be temperament unless they're tidal locked to their star. Then it's all about placement. The side that's locked toward the star would be hot, then a safe zone around the Dawn Belt and finally a frozen wasteland on the other side. Our moon is Tidal Locked to the Earth meaning that for millions of years, we've only seen one side of her.

If you have a dimensional disturbance story (a.k.a magic) or a possible Fey Loci then that would solve the problem as well.