Definition: A metric space is a set $X$ together with a distance function $d(x,y)$ that assigns a nonnegative number to any pair of points $x$ and $y$ from $X$, and $d$ satisfies:
- $d(x,y) = 0 \iff x=y $.
- $d(x,y)=d(y,x)$, and
- $d(x,y) \leq d(x,z) + d(z,y)$ ("the triangle inequality").
Definition: for a subset $A \subset X$,
$$ \operatorname{diam} A := \sup \{d(x,y): x \in A, y \in A \} .$$
Definition: for a point $ x \in X $, and $r>0$, the ball of radius $r$ centered at is
$$ B(x,r): = \{ y\in X: d(x,y) < r \}.$$
The story of the Hausdorff measure/dimension begins by extending our Euclidean lengths, areas, and volumes to arbitrary dimensions, in a rather crude way: in Euclidean $\mathbb{R}^n$, the $n$-dimensional volume of a ball is
$$ \omega_n r^n,$$
where $\omega_n$ is a constant independent of radius. Notice that if radius is scaled by $c>0$ the volume scales by $c^n$. The numbers $\omega_n$ are the all famous ones in dimensions 1, 2, and 3 and for the higher dimensions $\omega_n$ is obtained from $\omega_{n-1}$, recursively, by applying Fubini's theorem: a slice of $n$-sphere is an $(n-1)$-sphere with radius depending on the height at which we slice. Think about cutting a potato, a 3-ball, and getting (2D) disks!
It follows also that $\omega_n$ has an explicit formula via the Gamma function. Therefore, one can plug in noninteger values in place of $n$'s and (formally) define $\omega_\alpha$.
(In what follows $\alpha \in [0,\infty)$.)
Assume you have a metric space and you suspect that it is $\alpha$-dimensional. Then you should assign an $\alpha$-dimensional volume of
$$ \omega_\alpha r^\alpha ,$$
to the balls of radius $r$, i.e $B(x,r)$.
And a subset $A$ compared to a ball of same diameter has volume "comparable" (no precise argument here) to
$$ \omega_\alpha (\frac{\text{diam} \ A}{2})^\alpha.$$
Now, given a subset $E\subset X$, to define its $\alpha$-dimensional Hausdorff measure, $H^\alpha(E)$, we first find for a $\delta >0$ the quantity
$$ H^\alpha_\delta (E) = \inf \sum_i \omega_\alpha (\frac{\text{diam} \ A_i}{2})^\alpha,$$
where infimum is over all countable coverings $\cup_{i=1}^\infty A_i \supset E$ with no restrictions on $A_i$'s other than $\operatorname{diam} A_i < \delta$ for each $i$. That is, we look for the tightest coverings with sets of diameter less than $\delta$.
(In the definitions of $H^\alpha_\delta$ and $H^\alpha$ we allow infinities.)
Now,
$$ H^\alpha(E) = \lim_{\delta \to 0} H^\alpha_\delta(E).$$
Notice that smaller $\delta$ makes $H^\alpha_\delta$ larger, since we do not allow some of the coverings that previously worked, hence the infimum goes up. So, this limit exists since $H^\alpha_\delta$ increases as $\delta \to 0$.
With all this buildup we are ready to harvest the fruits!
Lemma: Given any metric space $(X,d)$, any subset $A \subset X$, there exists an $\alpha_0 \in [0,\infty)$ such that
\begin{align}
H^\alpha(A) &= 0 &\forall \alpha > \alpha_0, \\
H^\alpha(A) &= \infty &\forall \alpha < \alpha_0.
\end{align}
We call this $\alpha_0$ the Hausdorff dimension of the (sub)set $A$. The theorem tells us that too large measuring cups will not detect $A$ and smaller measuring cups won't contain $A$, which is quite intuitive and geometric: the area of a line is zero, and length of a (solid/filled) square is infinite.
This is how and why of a dimension of $1.25$ for the coast of Britain—this is the right $\alpha_0$ for it. Now, you ask then how much is this measure? This is the same as asking what happens right at $\alpha = \alpha_0$, which the lemma above leaves out?
There are three possibilities: $H^\alpha(E) = 0$, or $=\infty$, or it is a finite positive number. In each case the value deserves to be called the $\alpha$-dimensional measure/volume of $E$. Easy examples can be given for all cases. For instance, although a plane, extended indefinitely, is 2D, its 2D area is infinity nevertheless.
With specific and explicit metric spaces, specially those constructed through self-similar processes, it is usually straightforward to find the dimension—using the steps of the construction to find coverings that apply to the definition of $H^\alpha_\delta$. But finding the actual value $H^\alpha(E)$, may prove more difficult.
I do not know if this counts as "physical meaning" to non-integer Hausdorff dimensions and measures, but to me it is so natural and geometric!
