I absolutely agree with the other answers declaring this to be unsurvivable. But I'm going to try and make the case it could be survivable, just because I don't think anyone has tried sufficiently hard at thinking of ways it could be done.
Firstly, everyone has assumed the planets would inevitably approach at interplanetary collision velocities. This does not have to be true! If we're going to give ourselves the best chance of surviving, then we have a bit of preparatory engineering work to do.
So, we start with the planets orbiting each other, in a binary Klemperer rosette. Then, we induce the collision by gradually slowing them down, so that they gradually approach each other, until they just touch - still orbiting around their barycenter. Their relative velocity at impact is zero.
To minimize disruption at the area of impact, they'd have to be tidally locked, too, so their surfaces don't scrape across each other.
Exactly how to reduce their orbital velocities with such finesse is unspecified. Perhaps you can paint one half of the planet white and the other black, then wait a long time? There are fictional precedents of such planetary engineering feats, but they involved a reaction-less, inertia-less drive.
Without any orbital velocity, the planets would messily coalesce, churning crusts and mantles and cores, ejecting huge chunks into space to rain down afterwards, shedding every last whisp of oceans and atmosphere.
But with the orbital velocity, that can't happen. The planets remain suspended, each looming across half a sky. They won't remain the same shape of oblate spheroid as they originally were, of course. They'll flex and buckle, thrusting parts of the planet clear out of the atmosphere, while other parts drown in oceans of magma, hundreds of miles deep. God's own storms as half the atmosphere slews off into space. The sloshing oceans scraping a mile-deep layer off the crust as they go. Tidal forces yawning open all the old tectonic seams, and popping a hundred new ones too, as the rocks liquify under local strain variation.
And yet... somehow, the result is kind of stable. Not geologically, maybe. But for hours, or maybe years, they remain, a curious whirling hourglass of two kissing spheres. Long enough for someone who somehow predicted which small patch of crust would remain intact, to stand on it, in a space-suit, in a bunker, and say "I survived the event", before getting picked up and flown the heck out of dodge. Just before the 500 mile wave of ferociously radioactive liquid iron spurts out from the core, taking everything in its path. Who can say?
I'd originally planned to speculate about thousands of survival capsules spread around the planet, each equipped with miraculous ways of deriving power, and oxygen and water and... crap, what are they going to do for food? No, that's not going to work. Long term survival is, I concede, impossible.
Update: I concede the idea from other answers that, unless the planets were made of something unbelievably strong, tidal forces as they approached would flex and destroy their spherical shapes, generating tremendous heat, and tearing apart the planets into giant smears of molten rock. This would happen long before the planet surfaces could touch. So this answer is a nice poetic idea, but I don't believe it could happen after all.