This is inspired partly by this question, especially Tom Leinster's answer.
Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who probably don't much about the following.
Classically, there are two obvious categories whose objects are Banach spaces (let's say all Banach spaces are real for simplicity): in the first, morphisms are bounded linear maps and isomorphisms are exactly what are usually called isomorphisms of Banach spaces; in the second, morphisms are linear contractions (bounded linear maps with norm at most 1) and isomorphisms are linear isometries. These categories distinguish between the "isomorphic" and "isometric" theories of Banach spaces.
Now if I'm interested in finite-dimensional spaces, then the "isomorphic" category is not rigid enough, because any two n-dimensional Banach spaces are isomorphic. But the isometric category is too rigid for most purposes. So we get more quantitative about our isomorphisms. One way to do this is with the Banach-Mazur distance. If X and Y are both n-dimensional Banach spaces,
$$d(X,Y) = \inf_T (\lVert T \rVert \lVert T^{-1} \rVert),$$
where the infimum is over all linear isomorphisms $T:X\to Y$. Then $\log d$ is a metric on the class of isometry classes of n-dimensional Banach spaces.
Theorems about spaces of arbitrary dimension which include some constants independent of the dimension are characterized as "isomorphic results". One example is Kashin's theorem: There exists a constant c>0 such that for every n, $\ell_1^n$ has a subspace X with $\dim X = m= \lfloor n/2 \rfloor$ such that $d(X,\ell_2^m) < c$. (Here $\ell_p^n$ denotes R^n with the $\ell_p$ norm $\lVert x \rVert_p = (\sum |x_i|^p)^{1/p}$.) Thus $\ell_1^n$ contains an n/2-dimensional subspace isomorphic to Hilbert space in a dimension-independent way.
On the other hand there are "almost isometric" results, typified by Dvoretzky's theorem: There exists a function $f$ such that for every n-dimensional Banach space X and every $\varepsilon > 0$, X has a subspace Y with $\dim Y = m \ge f(\varepsilon) \log (n+1)$ such that $d(Y,\ell_2^m) < 1+\varepsilon$. Thus any space contains subspaces, of not too small dimension, which are arbitrarily close to being isometrically Hilbert spaces.
So my question, finally, is: are there natural categories in which to interpret such results? I suppose that the objects should not be individual spaces, but sequences of spaces with increasing dimensions. In particular, as the two results quoted above highlight, the sequence of n-dimensional Hilbert spaces $\ell_2^n$ should play a distinguished role. But I have no idea what the morphisms should be to accomodate quantitative control over norms in these ways.
Best Answer
This is a rewording of Matthew's answer, but one can use nonstandard analysis to obtain a category that achieves what Mark wants. After taking ultrapowers, one obtains non-standard integers (ultralimits of standard integers), nonstandard finite dimensional Banach spaces (ultralimits of standard finite dimensional Banach spaces), nonstandard linear transformations (ultralimits of standard linear transformations), etc. One can then separate these non-standard objects into various classes, e.g. bounded non-standard reals (ultralimits of uniformly bounded standard reals, or equivalently non-standard reals that are bounded in magnitude by a standard real), bounded nonstandard linear transformations, poly(n)-bounded nonstandard linear transformations, polylog(n)-bounded nonstandard linear transformations, etc., where n is an unbounded nonstandard natural number (which, in practice, would be used to bound dimensions of things). Each of these form a category.
Thus, for instance Kashin's theorem becomes the statement that every nonstandard Banach space of some nonstandard finite dimension N has an M-dimensional subspace which is isomorphic (in the category of bounded nonstandard linear transformations) to $\ell^2(M)$ whenever $M \leq N/2$ (or more generally when $M \leq (1-\epsilon)N$ for some standard $\epsilon > 0$.
Dvoretsky's theorem is trickier. Here, I guess one needs to work with the category of almost contractions: operators whose operator norm is at most $1+o(1)$ (i.e. bounded by $1+\epsilon$ for every standard $\epsilon > 0$. Then I think the theorem says that any nonstandard finite dimensional Banach space with some nonstandard dimension N has a M-dimensional subspace that is almost isometric to $\ell^2(M)$, whenever $M = o(\log N)$. (I may have messed up the quantifiers slightly, but this is pretty close to what the nonstandard translation of things should be.)