How far is the horizon?

On Earth, assuming Earth is a perfect featureless sphere (obvs, it ain't), it's about 5KM away from my vantage point 6ft above the ground. 1000ft above the ground, the horizon would be about 39 miles away; 47.5 miles if you were 1500ft away.

If you want to be able to see 100 miles across the sea, then I'm afraid your castle will need to be about 7000 feet up.

It will be different if the planet you are setting it on is larger or smaller, of course.

Essentially you're working out the length of the third side of a right-angled triangle, where the hypotenuse is (radius of planet + height of observer) and the other known side is (radius of planet) - just use Pythagoras.

I asked this question a few years ago, and got some really good replies. Post #4 in that thread provides a link to a calculator. ETA; Aw nuts, the link goes to a different website now.

Torgo's right, that your castle would have to be impossibly tall to see 100 miles. If it's SF/F, you might have to either be on a much bigger world, with less than the "8 inches to the mile" curvature, or have a seer who can see that far. As people pointed out in my thread, humidity and such will cause things to become hazy the farther you are trying to look.

It isn't quite that easy. Torgo is correct -- for a height of eye of 6 feet, the horizon (over water) is 3 miles away. The formula is:
D=1.22 * (square root of h) where D is distance in miles and h is height of eye in feet.
However, the OP is looking to see how far a distant land is given that it appears to be a thin grey line. To calculate this, you add the number you already calculated to the distance that an observer on the distant land could see the horizon.
So, D[SUB]2[/SUB]=1.22*(sqrt h[SUB]1[/SUB]+sqrt h[SUB]2[/SUB]) where D[SUB]2[/SUB] is distance in miles between the two observers, h[SUB]1[/SUB] is the height of the first observer in feet, and h[SUB]2[/SUB] is the height of the second observer in feet.
Therefore, given a spherical earth, and two 100 foot towers, to just see the top of one from the other, they would be 24.4 miles apart.
If you have a mountain range on the distant land that's 5,000 feet high, then you only need a tower 100 feet high to be able to see those mountain tops 100 miles away.

edit to add calculator link

Castles were often built on high points, so don't forget to take into account the height of the hill/mountain as well as the actual building.

In addition, the land that you sight could be mountainous and a couple of thousand feet above sea level.

Another factor what can be seen in the distance is refraction of light by the atmosphere. The exact effect is variable with altitudes and water vapor being the significant factors. On a humid Summer day the effect is significant and might increase the distance to the visible horizon by many miles. I have heard of people claiming to have seen things that were about 100 miles away from a height that is about 2500 feet.