To the extent of my creativity, L2 is the only location that can satisfy this for a given planet. We would also need to constrain the discussion to planets because, again, my creativity fails me to think of any other option.
Thus, we must use the method described in the other answer to ask "does any planetary L2 fall within the planet's umbra?"
To start with, the location of the L2 point can be approximated by the following, where M1 is the sun's mass and M2 is the planet's mass. R is the distance of the planet from the sun. r is the distance of the L2 point from the planet.
$$ M_2 \ll M_1 \\
r \approx R \sqrt[3]{\frac{M_2}{3 M_1}} $$
So for L2 to be close to the planet, we would want the planet's mass to be small. However, a smaller planet has a smaller radius. Let's move on to calculating the termination point of the umbra. This can be done with geometry, using similar triangles defined by the end of the umbra with the planet and with the sun.
$$ r_u \approx R \frac{ D_{planet} }{ D_{sun} } $$
Our basic criteria for an eternally shaded L2 point is then simple, that $r>r_u$ in the above equations. Here are my calculations of the ratio between the two values using planetary data:
- MERCURY 0.920628441
- VENUS 0.931131053
- EARTH 0.916884469
- MARS 1.02662275
- JUPITER 1.505844946
- SATURN 1.897836869
- URANUS 1.505423974
- NEPTUNE 1.382215047
- PLUTO 1.322002729
For all values above 1, the L2 point is fully shaded. If the value is very close to 1, there are likely to be some other complications, since you might not be in exactly that point, so some sunlight will still get through. You'll also still need some station-keeping at that point to hold in place. Also, note that a space elevator is possible (in theory) for a point like the sun-Mercury L2 point, but the counter-weight would be further away than the L2 point itself, meaning that it would not be perfectly shaded.
I repeated these calculations for planet's perihelion and aphelion. They were exactly the same, so there must be some definitional reason why (unless I messed it up). But managing the L2 point within an elliptical orbit is a complicated maneuver, so this couldn't be so simple in real life.
But otherwise, the conclusion is pretty clear. Once you get out of the inner solar system, the sun-planet L2 points lie fully within the planet's umbra. I lack a general technical explanation for why this should be. I just used the data.