Yes, it is possible. For example Kevin Brown did here and here including this table.
so for the xkcd problem the answer is $-\frac{1}{2}+\frac{4}{\pi} \approx 0.773$.
The total resistance of the grid is infinite when the grid is two dimensional and large.
If you place two point probes at location x and y on an infinite 2-d resistor grid, and impose the voltage V(x)=1 and V(y)=0, the potential obeys the discretized Laplace equation: V(up) + V(down) + V(left) + V(right) - 4 V(center) = 0 with the boundary conditions at the two given points and V=0 at infinity (beyond x and y).
In the limit that x and y are far apart, the discrete Laplace equation might as well be the continuous Laplace equation, and the solution goes like C log(|r-x|/|y-x|), so that the potential difference for any finite C diverges with the distance. This means that C has to go to zero in the large |x-y| limit, so the current vanishes. The same is true in 1d, where a line of resistors has a current which vanishes as 1/L, so the total resistance goes as the total length L. In two dimensions, the total resistance blows up as log(L).
For a three dimensional grid and higher, you do have a finite resistance for a block. Whether the limiting resistance is finite or infinite is the same problem as the recurrence/nonrecurrence of a random walk on the grid.
If you make a pseudo two-d grid using N parallel lines of N resistors in series, then the total resistance is N on each path, but there are N parallel paths, so the total resistance is R, independent of the size. This is not the same as the 2-d resistor grid, because in the 2d grid there is resistance to going vertically a long way which is similar to the resistance to going horizontally, so the horizontal resistor paths are not parallel. If you make all the vertical resistors zero, and make the separation between x and y horizontal, and make the vertical width equal to |x-y|, you recover the series/parallel situation.
The series-parallel example gives intuition about why two dimensions is critical for the transition from infinite resistance to finite resistance.
Best Answer
A good reference was given in an answer to a related question: Cserti 2000 (arXiv preprint, whose numbers I'll be referring to) solved a number of generalizations of the 2D lattice problem.
For a $d$-dimensional lattice, the resistance between the origin and the point $(l_1, \ldots, l_d)$ is given by eq. 18 in that paper: $$ R(l_1, \ldots, l_d) = R_0 \int_{-\pi}^\pi \frac{\mathrm{d}x_1}{2\pi} \cdots \int_{-\pi}^\pi \frac{\mathrm{d}x_d}{2\pi} \left(1 - \mathrm{e}^{\mathrm{i}(l_1x_1+\cdots+l_dx_d)}\right) \left(\sum_{i=1}^d (1 - \cos x_i)\right)^{-1}, $$ where $R_0$ is the resistance of a single resistor.
The author goes on to talk about the specific case of $d = 3$, where in eq. 35 it is shown that $$ R(l_1, l_2, l_3) = R_0 \int_{-\pi}^\pi \frac{\mathrm{d}x_1}{2\pi} \int_{-\pi}^\pi \frac{\mathrm{d}x_2}{2\pi} \int_{-\pi}^\pi \frac{\mathrm{d}x_3}{2\pi} \frac{1-\cos(l_1x_1+l_2x_2+l_3x_3)}{3-\cos x_1-\cos x_2-\cos x_3}. $$ Moreover, as one might expect with enough dimensions, the increase in the number of paths between points can grow fast enough to balance the increase in path length. Indeed, for $d = 3$ the resistance between two points asymptotes to a constant as the points diverge. The paper gives the constant in several forms, one of which is eq. 39: $$ R_\infty = \frac{\sqrt{3}-1}{96\pi^3} \Gamma^2\left(\frac{1}{24}\right) \Gamma^2\left(\frac{11}{24}\right) R_0 \approx 0.505 R_0. $$ Note that this means even if we stick to a single plane in a 3D lattice, the existence of the third dimension drastically alters the qualitative behavior as we move the two nodes apart, since in 2D $R_\infty$ diverges.