Well, no simple analog circuit is going to give you 2nd harmonic distortion ONLY. Tube amps do not do this.
What you probably want is a circuit that generates EVEN harmonics - not just the 2nd harmonic.
(Note that, apart from the technical difficulty, distorting the input signal to generate the 2nd harmonic ONLY would probably sound rather uninteresting on a guitar signal.)
Basically, even harmonics appear when the signal loses its symmetry. So what you want is a circuit that will "amplify" (quotes because it doesn't need to have a gain > 1 for this) the signal asymmetrically.
Additionally, you usually want to not just generate even harmonics, but minimize the odd harmonics too. For this, a "soft clipping" behavior helps.
So, all in all, you want a combination of both: asymmetric amplification and soft clipping. As some have suggested above, a simple, yet working technique is what has been done in "overdrive" pedals for a long time: opamp-based circuits with diodes in the feedback path.
For asymmetry, you can use a different number of diodes in each direction, as also suggested above. Or, you can use the same number of diodes in each direction, and add some offset voltage to the input signal. It'll get you asymmetry as well. (Of course don't forget to add a final series capacitor at your circuit's output to remove DC.) You can use a combination of both. You can also cascade stages like this. You'll also need to add filters. IIRC, the Boss OD has an high-pass filter before the saturation stage, and a low-pass filter after. This is because distorting equally low and high frequencies of your input signal would yield a pretty "muddy" sound. The final low-pass is to limit the high-frequency components after the saturation stage (there can be a lot of harmonics in the output, and even if they'll be mostly "even" harmonics, that won't sound pleasing at all if you don't cut them off to a reasonable frequency.)
In any case, you can look up schematics of the original Boss OD: they are easily found on the web.