Science

(Caveat: this is kinda outside my normal field, some part of this is probably wrong or misleading in some way.)

This is the wrong way to think about this, for kind of a subtle reason. So generally when we talk about dimensions in science we mean one of two things:

• The kind of units something has (so if a quantity can be measured in seconds/years/etc it has time dimensions, if it can me measured in meters/inches/etc it has length dimensions, and so on). That's clearly not the kind of dimension you're asking about.

• How many numbers you need to describe a particular system (so a particle in space needs three dimensions to describe it's location – or four if we want to consider time – and we can approximate the position of a pendulum in a clock to a system that can be described in one dimension: how far along we are in one swing cycle). I believe you're talking about this kind of dimension: extra directions that we could, in theory, travel in: ie, an extra fifth (sixth, seventh, eighth, etc) number beyond the normal four we have in "everyday" spacetime.

Consider a sphere. It's two dimensional and both dimensions are unbounded (no edges), finite, and curved (convex in this case). Now, you're probably imaging a sphere in 3D space, because that's what our brains are wired to do. But this is a mathematical sort of object, and it turns out that the third spatial dimension is completely unneeded: you only need two numbers to describe where you are on a sphere (for example, latitude and longitude). In this case, the sphere doesn't need to be contained in anything! We can just toss out that third number, it's an extra. The sphere just is.

Likewise, our four dimensions of spacetime don't need to be contained in anything else. Their topology is different, sure. The space dimensions are, as far as we can tell, unbounded (no edges), infinite, and basically flat; time is more complicated. But just like you can use three numbers to describe your position on a sphere but only need two, you can use a fifth number to describe your position in spacetime but it's an extra. And, just like the sphere, the dimensions don't need to be in anything, they just are.

Now, you could say: "What if we assume that we do actually need five numbers, and the four we normally use are actually just a simplification?" That's a valid thing to do (although hard or impossible to prove), but in that case you can play the same game by adding a sixth number. That number is extra and not actually needed to contain the five dimensions in our new theory.

So tl;dr dimensions don't need to be inside of anything. You can embed different kinds of spaces into higher dimensional ones by adding an extra number, but even in that case you don't need anything to "hold" the extra number. Dimensions just are.

I hope this is more helpful than it is confusing! (Physicists/mathematicians: feel free to correct me if I got anything egregiously wrong here.)