Here is a piecemeal post record some follow up to [John Rennie's answer](http://physics.stackexchange.com/a/298871/26076).

There are general embedding theorems for manifolds, which show that there is always a way to think of any manifold as an embedding in a higher dimensional, flat manifold. This is essentially what NDT is talking about. Although such embeddings are always in principle possible, there are limitations to these theorems that mean we sometimes cannot do them in a way that is useful for physics.

The most general and intuitive theorem is the [Strong Whitney Embedding theorem](https://en.wikipedia.org/wiki/Whitney_embedding_theorem) which shows that any real, smooth, Hausdorff and second countable m-dimensional manifold manifold can be embedded in $\mathbb{R}^n$, where $n$ is at most $2\,m$. 

Historically, this was important, because it showed that the modern conception of a manifold (as a geometrical object charted on overlapping patches) was the same notion as the dominant early conception of a manifold as simply some hypersurface in $\mathbb{R}^n$.

But this is not all we want for differential geometry General Relativity, of course. We are concerned with Riemannian and (for GR) Lorentzian manifolds, *i.e.* manifolds whose tangent spaces are kitted with a nondegenerate symmetric billinear form (inner product), whence one defines the concept of a metric (or pseudometric, in the case of Lorentzian manifolds).

So we want embeddings to preserve this structure on the tangent space. We want them to preserve length. Thus we are concerned with the much more specialized *isometric* embeddings. The Whitney theorem is not concerned with this further structure to work, so it is no surprise that the embeddings whose existence it guarantees may not be isometric ones.

Enter now the [Nash Embedding Theorem](https://en.wikipedia.org/wiki/Nash_embedding_theorem). This guarantees an isometric embedding of any  Riemannian manifold (*i.e.* if the symmetric billinear form is positive definite - which is NOT what we are concerned with in GR) into a higher dimension Euclidean space. If the manifold is compact, and of dimension $m$, it needs at most a Euclidean space of dimension $\frac{m}{2}(3\,m +11)$ to be embedded into. Things are even worse for a noncompact space: the bound is $\frac{m}{2}(m+1)(3\,m +11)$.

But this result does not work for a Lorentzian manifold, where the billinear form is not positive definite and there are null vectors and light cones. In particular, closed timelike curves destroy Nash's central argument.

There are weaker embedding theorems for pseudo-Riemannian manifolds, with nonpositive definite billinear form. See this [Math Overflow Thread](https://mathoverflow.net/questions/127734/nash-embedding-theorems-for-pseudo-riemannian-manifolds) for a discussion. They still guarantee an isometric embedding, but these theorems are weaker in the sense that, unlike the Nash theorem, there are no bounds on dimensions - they give no guarantee as to how many dimensions you need to embed in.

**There is one critical point that also prevents them from being generally useful in physics.**

For a Lorentzian manifold, the embedding theorems above tell you nothing about the number of *timelike dimensions*. Sure, you can always embed the Lorentzian model, but, in the words of ACuriousMind:

>*An isometric embedding in particular preserves the time/light/spacelike character of a curve, but there are no closed timelike curves in Minkowski space, so you simply can't have such an embedding without raising the number of temporal dimensions on the target space.*

That is, if you isometrically embed our signature $(1,\,3)$ manifold into a flat higher dimensional one, and if our manifold has closed timelike curves, then the embedding is going to be of the form  $(n,\,m)$ where $m\geq3$ and $n$ is *strictly greater than 1*. So we shall end up with more than one timelike dimension, and that is not generally useful for physics. Not the least because there are then always isometries on the metric that both belong to the identity component of the group of isometries AND reverse the time direction of a vector. So it becomes very hard, if not impossible, to do any physics with a notion of causality in such an embedding.

If, however, the manifold is [globally hyperbolic](https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold), then the Clarke embedding theorem guarantees an isometric embedding with one timelike dimension, see e.g. theorem 2.8 [here](http://www.damtp.cam.ac.uk/user/st457/essay.pdf).