Can a ball of yarn (seen by an observer with limited resolution) be correctly described as a manifold? The idea of box-counting dimensions is that the dimensionality of an object can be (somewhat) thought of as dependent on the resolution of the observer. 
If I think of the side length of the boxes necessary to cover a figure as a type of "resolution limit", than I imagine that dimensionality of some objects (like a ball of yarn) will vary from one observer to another. Large boxes would imply a very course resolution, while small boxes would imply very fine resolution. From this point of view, the dimensionality of a ball of yarn is $1≤d≤3$ (e.g. the the ball of yarn appears as a 1D object if my resolution is very good, but will appear as a normal 3D ball if my resolution is poor (to the point where I can't discern the individual strings)). (e.g. Every experiment available to that observer will give a result consistent with the ball of yarn being just a 3D ball.)
{\bf Can an observer with low resolution (correctly) claim that a ball of yarn is a 3D manifold? From our view, we know that that the yarn is fundamentally a wound up 1D object, and that the observer's claim of the ball of yarn being 3D is due to the fact that the observer cannot resolve the individual strings? If the yarn is not a manifold, is there a type or class of 'manifold-like objects' that this might resemble? 
Can one write a metric or metric-like object (since 'metric' does have a formal definition) where the number of spacial dimensions can be observe-dependent?
A somewhat follow-up question (more physics-centered, so I don't know if it will be answered here) is: can someone then write a theory of gravity (with a Riemann curvature tensor, Ricci Tensor, Christoffel Connections, etc. (or equivalent since this is an unusual scenario)) on a background similar to a ball of yarn which contains a resolution/scale-dependent dimensionality. I imagine it should be possible since in a sense this is the opposite of string theory.
(I'm just going to throw out that I am not trained in mathematics, so I may struggle with formalisms.)
 A: The box-dimension of a fractal (also known as Minkowski dimension) may be defined by (there exist other, equivalent definitions):
$d_M(\mathcal{F}):=\lim_{\epsilon\rightarrow 0}\frac{\ln(N(\epsilon))}{\ln(1/\epsilon)}$, where $N(\epsilon)$ denotes the number of boxes of sidelength $\epsilon$ to cover the fractal $\mathcal{F}$. (Note that if this limit does not exist, you can nevertheless define upper and lower Minkowski dimension by using upper and lower limit.)
Thus your question does not really make sense from the mathematical point of view. However I think I understand what you mean. You are probably thinking of a fractal analogue to the process/technique known as compactification in string theory. This is an extremely interesting question, but to my knowledge, there does not yet exist a viable formulation yet. You may find a few informations in some papers published by K. Svozil from Vienna University of Technology. I can't give you the precise references, as it is quite some years ago when I read them, but maybe you will be able to track them down. There is also a book by L. Nottale called Fractal Space-Time and Micro-physics, but I am sceptical about the scientific quality of his approach.
Maybe this is not the answer you were looking for but it is always a pleasure to read about less "mainstream" ideas.
A: For the most generalized definition of "metric" (a metric space), there is no reason why it cannot be resolution-dependent. A quick crack at it would be the following:
For every $\epsilon > 0$, you have a set $X_\epsilon$, and a function $d_\epsilon: X_\epsilon \times X_\epsilon \to \mathbb{R}$ which satisfies the following three properties for all $x,y,z \in X_\epsilon$:


*

*$d_\epsilon(x,y) \ge 0$ (Non-negativity)

*$d_\epsilon(x,y) = 0$ if and only if $x = y$ (Identity of Indiscernibles)

*$d_\epsilon(x,y) = d_\epsilon(y,x)$ (Symmetry)

*$d_\epsilon(x,z) \le d_\epsilon(x,y) + d_\epsilon(y,z)$ (Triangle inequality)


You would then want a compatibility condition between the different metric spaces $(X_\epsilon, d_\epsilon)$. An idea would be to require for all $\delta < \epsilon$, $X_\delta \subseteq X_\epsilon$, and that $\lim_{\epsilon \to 0} X_\epsilon = B$, where $B$ is your ball of yarn. 
Since the $X_\epsilon$ are simply sets, there is no definite dimension to them, and so the dimension (how ever it is defined) can change with $\epsilon$. 
You can apply this definition to your example by letting the $X_\epsilon$ be the union of the boxes with side length $\epsilon$, and letting $d_\epsilon(x,y)$ be the length of the shortest path from $x$ to $y$ within the boxes. As $\epsilon \to 0$, the metric space so constructed would be very similar to the metric on (a connected, bounded subspace of) $\mathbb{R}$, while when $\epsilon$ is large, the metric space will be similar to the metric on (a connected, bounded subspace of) $\mathbb{R}^3$. You can make this rigorous with a metric-preserving map from $(X_\epsilon, d_\epsilon)$ to $\mathbb{R}^n$ where $n = 1$ or $3$.
If you take this idea further, you can probably use it to construct a resolution-dependent manifold from the $X_\epsilon$. I, however, am lacking in the expertise necessary for that challenge.
A: I would attempt to give an answer to the last question/ follow up question.
Dimensionality (and topology, I mention it, because it behaves similarly in some sense), can be thought of as "effective" quantities, especially if one talks about models of gravity / quantum gravity. There is no reason to believe, that our spacetime is 4 dimensional at start (even though right now it seems so), an that it has spherical or whatever other funny topology. On the level of the Planck scale or in cosmological scales the observations of dimensionality can differ from the "classical picture". A phenomena that captures it is called "dimensional reduction".
This phenomena appears in many models of quantum gravity (including Euclidean- or Causal- Dynamical Triangulations, Loop Quantum Gravity, Causal Sets, Group Field Theories...), which analyses the dimensionality of spacetime from the perspective of an observer. That is true that the Box counting dimension or the Hausdorff dimension will give 4 in those models, however if you define the dimensionality as a quantity related to the return probability of a diffusion, then you can define the so called "spectral dimension". Lets imagine a diffuson in a $d$ dimensional space with the heat kernel $K$
$$K(x,x_0,\tau = 0) = \delta^d(x-x0) $$, where $\delta$ is the Dirac delta.
If the total probability is preserved as:
$$\int d^dx \sqrt{g}K(x,x_0,\tau) = 1$$, then the average return probability to the origin can be given by:
$$ P(\sigma) = \frac{1}{V}\int dx \sqrt{g}K(x,x,\tau) $$, where $V$ is the volume of the underlying $\mathcal{M}$ manifold.
The spectral dimension then $d_S$ is given by:
$$d_S = -2\frac{dlog(P(\tau))}{dlog(\tau)}$$.
The name spectral dimension comes from the fact, that the operator driving the diffusion is the Laplace-Beltrami operator, and its eigenvalues can be also related to the return probability:
$$ P(\tau) = \frac{1}{V}\sum_i e^{-\lambda_i \tau }$$, and thne taking the asymptotic expansion:
$$P(\tau) = \tau^{-\frac{d_s}{2}} \sum_i A_i \tau_i $$, with $A_i$ being some functions of the metric.
Surprisingly, $d_S$ is a quantity that strongly depends on the scale. On large scales this quantity gives back the large scale dimensionality, but on short scales due to the "fractalness" of the spacetime it gives a smaller value (generally accepted is $d_s = 2$, but some argues it is $1.5$).
Models, such as Causal Dynamical Triangulations finds it numerically, where starting from 4-dimensional simplices as building blocs one discretizes spacetime and write down the Einstein-Hilbert action in the formalism of Regge. Then using Monte Carlo simulations one may find phases, where the geometric properties of the spacetime strongly differ from each other. One characteristic phase is the "branched polymer" phase, with effective Hausdorff dimension $d_H = 2$ and short scale $d_S = 4/3$. Even though the numerical constraints force the "mathematical/numerical" topology and dimensionality to be equal with the initial values, the observables "existing" in the triangulation would observe an effective behaviour differing from that. There is another phase, called deSitter phase, where the spacetime, which is initially has $T^1 \times S^3$ topology ( where there is a globally hyperbolic foliation of 3-spheres winding around in the time direction), transforms into an $S^4$ in the "effective" description, yielding $d_H = 4$ but $d_S = 2$ at short scales.
As it is a theory of quantum gravity, this result means, that spacetime on the shortest scales could potentially behave as a lower dimensional manifold, even though the large scale properties are different.
This also means, that the dispersion relations of scalar, vector, tensor quantities do not have to agree with each other (as they would do in the classical level), but some could observe different dimensionality on different scales.
