Dumbbell planet - how would humans cross the bridge?

Note: I am aware of this question Could two planets be tidally locked to each other so close they share their atmosphere? My question is not about that. It is specifically about human travel.

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By a cosmic accident, Slartibartfast's design, or through God's will there is a planet that is shaped like an old-fashioned dumbbell.

Each lobe is Earth-sized and Earth-like in terms of their basic composition - at least on the surface. What lies under the surface may be assumed to be whatever is necessary to hold the structure together. You can assume a degree of hollowness if that helps with the plausibility.

The 'bar' joining the lobes is mainly a natural form of steel. It connects directly to the core of each lobe. Its proportions are exactly as shown in the picture. The bar is the same length as the diameter of the lobes.

Question

Human life develops on one lobe, either by evolution or by being created.

Under the normal laws of physics:

Would the humans be able to walk from one lobe to the other or would it be like climbing and then descending an enormous mountain? If they can't simply walk - what stage of technology would they need in order to cross the bridge? How would they actually do it?

Notes

1. Please feel free to discuss factors like: shared atmosphere, the most stable axis of rotation, the effect on seasons, on tides and on day and night etc. However my prime interest is whether humans can cross the bridge and what level of technology it would take.

2. If you need to presume a particular internal structure for the planet to make it more plausible, then please do so and say how that will affect the answer to the travel problem.

EDIT

The answers so far (29/07/15) are very useful. However I feel that maybe not enough attention has been given to the gravitational pull of the bridge itself. If this picture is accurate with regard to relative size of Earth and Moon, then I imagine the gravitation due to the bridge will be at least equal to that of the moon - probably a lot more. It's certainly not negligible. This would mean that, at the half-way point between lobes, humans could indeed walk around without floating off. Presumably they could colonise the area. Maybe they could mine it and simply throw the product down towards either side. I imagine they could use it as a platform for space-travel.

Wouldn't the proximity of the two lobes, combined with the gravitational pull of the bridge mean that it could maintain at least a tenuous atmosphere?

UPDATE

Thanks to comprehensive and convincing answers so far, I am now resigned to using large amounts of unobtanium in the actual construction of the planet.

The joining rod is coated in steel that has magnetic properties but the steel is coated with thick flaking rust. The surface underneath is pockmarked to some degree according to the weather at any give altitude. The bar is supported on the inside with unobtainium scaffolding. This is 'plated' with 1000 metres thickness of steel. In theory one could drill through to the hollow interior. There are holes in the unobtainium scaffolding that are big enough to allow very large items of machinery to be introduced once the steel has been broached (perhaps even as big as a moderate sized trawler).

My request now is to provide the earliest possible technology (by our Earth date) that would allow travel between lobes.

Clearly hot air balloons wouldn't come anywhere near, whereas rocket science would suffice. The question is, How far back in standard Earth history could we go and still be able to make the journey?

Examples

If you suggest welding, then please give evidence of when the necessary sort of welding was invented.

If you suggest magnetic vehicles, please say when magnets would have been strong enough for the purpose.

If you suggest breathing apparatus, please say when it was invented or at least could have been invented with materials available at the time.

If stone-age man could have made breathing apparatus out of flint (unlikely!) then that is okay.

The winner will be whoever comes back offers and substantiates the the oldest possible technology.

Please feel free to amend current answers or add new ones.

Good luck and thank you.