First, we talk about area of shape on Euclidean plane:
Back in ancient time, Greek (such as Euclid) do not truly assign numerical real value as area of shape, unlike us today. In fact, object have same area if we can cut them up into finite piece and use isometry (reflect, rotate, translate) to turn one into another. That is minimally good enough, because all we care about is that what ever algebraic structure that contains all possible value for area to have the operation of addition and a total ordering. So strictly, we do not truly need measure theory (as it restrict you to just real value), and of course, measure theorem have COUNTABLE additivity, which can be considered too strong a requirement. A modern theorem (Bolyai-Gerwien theorem) told us that given any 2 polygonal shape of the same conventional area, you can indeed always cut them into finite piece and use isometry to convert one into another. This validate Euclidean approach to defining area.
In modern time, we think of "shape" as more like set of points, and we want to assign numerical value as area to all subset, satisfying certain condition such as being invariant under some transformation, being finite additive, monotone, and a some normalization. This is already good enough for Archimedes to approximate $\pi$, without having ever deal with an infinite sum.
In 2-dimension, von Neumann showed that a square can be decomposed, apply area-preserving affine transformation to, and get back 2 square of the same size instead. However, luckily, such paradox cannot happen on polygonal shape and circular disk if only rigid motion are allowed, thanks to a theorem due to Banach. In fact, we can construct an assignment of area to all subset that is invariant under rigid motion and satisfy all other reasonable definition of area.
Now, to the question of area in higher dimension:
Hausdorff paradoxical decomposition of spherical shell (very related to the infamous Banach-Tarski paradox) means that we cannot hope to assign surface area to all possible subset that is invariant under rotation. Though of course, it's intuitively clear before that, because if a subset is too badly looking, there is no sense to talk about area because we don't even know what is its "surface".
Attempt to define it based on analogy with curve turn out to be a failure. A standard definition of length of curve is by using approximation with piecewise linear curve, and find the supremum of such approximation. It's a good approach for curve, as it can measure all sort of curve, unless of course if the curve behave so badly that we would intuitively assign a length of $\infty$ to it anyway. That approach turn out to be a failure for area on surface with curvature: you would get infinite area for a simple surface such as a cylinder (cylinder area paradox).
However, for nicest kind of area, the one on a differentiable manifold, we could in fact define area using analogy from curve, but this time analogy with differentiable curve. Beyond that, things get fuzzy, with various definition disagree with each other.
In conclusion, we don't really know once it comes to higher dimension. But we do have a consistent definition for area on 2 dimensional Euclidean plane.