Are neutral wires linked to the neutral bar in the fuse box safe to touch
If and only if everything is working

For a theoretical panel, neutral is theoretically safe to touch if: 


the panel is wired to modern spec
the panel is wired correctly 
workmanship is all proper
no wire/hardware failures (note: this is a different thing than "wired correctly") 
The supply transformer is isolating, and not leaking enough 9600V primary current to override the transformer's grounding rod (dirt isn't much of a conductor, that's why we don't just wrap it in insulation and call it wire)
you are not in the Philippines 


But then, if all that was true, you wouldn't be opening up your panel, would you? :)

But yes.  If things are in order, there will be a grounding electrode system going from ground rods, metal water pipe or Ufer tie-in to your basement's reinforcing rod, which establishes contact with earth proper.  There will be one at the transformer tying to neutral, and one at each building's most main panel tying to ground.    

Next, in every building the grounding electrode system will be bridged to the service-wide network of green, green/yellow or bare wire, or metal conduit, known as the equipment grounding conductor, or colloquially "ground" in the electrician's context, not to be confused with GND or Vss in electronics, which is why I steer out of my way to call this equipment safety ground.  No current ever flows on this except during a fault condition. 

Finally, one place in every service (i.e point fed by an electric meter, customer demarcation point, etc.), typically at the service point, has a neutral-ground equipotential bond.  That has several jobs. Relevant to your question, it is to do exactly what you are concerned about: clamp neutral so it is near ground and cannot kill you.  No current sgould flow on it because no current should be on ground, and transformers shouldn't leak.  

By the way, the neutral-ground bond is a great place to put a clamp ammeter. Helps to wire your panel so that is easy.  

In a residential setup, you optimistically have several houses sharing a transformer, and the transformer and each house's neutral-ground bonds are all wrestling neutral down to safe voltages. In an industrial setup, that is not the case, and I got a real eye-opener when a transformer's only neutral-ground was not actually attached.  


  I was working on a branch circuit, and to make sure the breaker was off, I brushed hot across the edge of the steel junction box.  There was no blinding arc flash, but something else - the light changed subtly.  Doing it again, the circuit powered back up, and a fluorescent light came on.  How could that happen? If hot is grounded, neutral is --  I measured neutral-ground: 120V. Crossed wire, in my work!? No.  I checked other circuits, it's the whole main panel.  HMMM!  Where's that coming from!?  Well, after shutting off 1 circuit at a time, it finally extinguished.  Turns out an unrelated circuit had a bolted hot-ground fault, which was bonding hot pole L1 to ground, for lack of a neutral-ground bond to argue the other way.  Normally this would cause unlimited current in series from L1 via the breaker, L1 wire, fault, equipment safety ground, neutral-ground bond and back to neutral, and snap the breaker.  As it was, it only flowed current when a second ground fault occurs, which I induced by flashing my circuit's hot to ground.  


Neutral could have bitten me at any time, why didn't it? I like to credit good practices: I treat neutral and other ”known”-dead wires as if they're live.  The takeaway is this: 

Neutral is live until you have done a physical inspection that all the earthing hardware is in working order, and done an electrical test to confirm.