It's a tougher question than it looks, because there's actually a good reason for men to supress their emotions... however, the reason why a real man should do so often goes unsaid.
In a difficult situation (such as responding to a disaster), a real man needs to keep a cool head so that he can do what needs to be done. It makes sense for someone to be emotional or distraught at such a time, but a real man knows full well than standing there in shock helps no one (himself included). It's not that he doesn't feel emotions, but now isn't the time to deal with it.
After the crisis has passed, he should be free to let it all out. Bottling up emotions when you don't have to is bad for your health, it just raises your stress level (leading to a host of other problems). Real men don't deny the fact that they have feelings, and will are free to express them. However, they know when being emotional is not the right thing to do and will supress their feelings temporarily.
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To address the general question of "What is manliness?", I would say that it isn't anything which can be described as machismo. Characters like Kratos from the God of War games are not the epitome of manliness, they are brutish neanderthals. In fact, it can be argued that a berserker rage is not the preferred way of expressing your anger; the better way is to be deadly calm [http://tvtropes.org/pmwiki/pmwiki.php/Main/TranquilFury] (and if possible, with an icy glare [http://tvtropes.org/pmwiki/pmwiki.php/Main/DeathGlare]). Control over one's self is one of the traits of manliness, and giving into primal urges such as rage & lust all the time is evidence to the contrary... and I'm getting away from the point here.
What truly defines manliness is leadership.
That's good leadership, not just being in charge. A real man is an authority figure (or a father figure, you could say) that everyone around him can trust & respect, and he has earned those honestly. He isn't in charge because he has exerted his dominance over his subordinates, but because rather they support him and expect him to make the right decisions and do the right things for all who are in his charge.
To be a leader is a responsibility, not a priviledge. A king is a servant to all of his subjects, less he become a tyrant. This concept has been around for ages, usually in the form of "Noblesse Oblige" [http://en.wikipedia.org/wiki/Noblesse_oblige] (French for "nobility obliges") and is still used to briefly describe for how a leader should act. However, there is a more modern version which I'm certain you all should be familar with:
"With great power comes great responsibility."
Said in every single iteration of Spiderman by the same character: Uncle Ben. That is the definition of manliness; using your power wisely and responsibly, not abusing said power just because you have it.