Why should mass be attractive in nature? Why does a mass attract all the masses around it? Why should't it repel or just stay calm? Why should it be like that?
 A: Gravity may be treated as a quantum field theory. In this kind of theory, interactions are represented by  field correlations, more known as "virtual particles", "virtual gravitons" in the case of gravity.
The fact that two charges (more precisely, in the case of the gravitation,  $2$ positive energy  densities) attract each other is due to the sign associated to a quantity named propagator which describes these field correlations. The sign of the propagator depends on the relation between the Lagrangian  and an invariant LorentZ quantity, described below, with the constraint that the time derivative of the spatial components of the fields get a positive sign in the Lagrangian.
Take a metric $g_{ij} = Diag (1,-1,-1,-1)$. This metrics is used to raise or lower indices.  
With a scalar field (spin 0), the invariant lorentz quantity is $\partial_\mu \phi \partial^\mu \phi = (\partial_0 \phi)^2 - \sum\limits_{i=1}^3(\partial_i \phi)^2$, the Lagrangian is then $L = + \partial_\mu \phi \partial^\mu \phi $
With a spin one field (electromagnetic field), the invariant Lorentz quantity is proportionnal to  $(\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial^\mu A^\nu - \partial^\nu A^\mu) = - 2 \sum\limits_{i=1}^3 (\partial_0 A_i)^2 + ....$
The minus sign here comes from $g^{ii}=-1$, for $i=1;2,3$. We must have a positive sign for the time derivatives of the spatial components, so we have to put a minus sign, that is $L $ proportionnal to $- (\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial^\mu A^\nu - \partial^\nu A^\mu)$
With the graviton, which has spin $2$, and is representing gravity interactions, the interesting part of the invariant Lorentz quantity is : $(\partial_\mu h_{\nu \rho} \partial^\mu h^{\nu \rho})  = + \sum\limits_{i,j=1}^3 (\partial_0 h_{ij})^2 + ....$
The $+$ sign comes from : $g^{ii} g^{jj} = (-1)^2= +1$, so the Lagrangian is $L = \partial_\mu h_{\nu \rho} \partial_\mu h^{\nu \rho}$ + other terms
This game with the sign has a direct physical consequence. When you have a $+$ sign, charges of same nature attract, and when you have a $-$ sign, charges of same nature repell. So, positive energy densities ("masses") attract each other by gravitational interaction. 
This is the general idea. Well, I made some simplifications...  , for a complete discussion, see : Zee (Quantum Field Theory in a nutshell), Chapter 1.5.
A: The masses can't repel each other because gravity is mediated by a spin 2 field, and for spin 2 the force between charges of equal signs is attractive. See the question Why is gravitation force always attractive? for an explanation of this.
But it's impossible to say why the force can't be zero. Experiment shows that masses do attract each other, and General Relativity tells us how masses attract each other. But as to why this happens, that's essentially asking why the value of Newton's constant, $G$, isn't zero. There is no answer to this other than the obvious one that if $G$ was zero we wouldn't be here to measure it. We don't have any theory that predicts the value of $G$.
A: 
Why does a mass attract all the masses around it? 

Because this is what has been observed . No apples falling up have been observed. 

Why should't it repel 

Because no repulsion of masses has been observed up to now. There exist experiments at CERN where the gravitational behavior of antimatter is probed and maybe in the future there will be an observation that antimatter has garden variety attractive gravity.
Therefore the mathematical theories modeling the observation have only attractive gravity, as described in the other answers, by construction, to fit the observations so as to be able to predict new behaviors. The predictive behavior of gravitational theories is extremely accurate.

or just stay calm?

This would mean at best just a primordial  inactive soup of neutral objects, a gas, may be; no mass accumulations , no planets, stars etc .... This is contradicted by observations

Why should it be like that?

"Why" belongs to the philosophy discussions.  At most, in physics discussions it leads to the "anthropic principle"

The anthropic principle (from Greek anthropos, meaning "human") is the philosophical consideration that observations of the Universe must be compatible with the conscious and sapient life that observes it. Some proponents of the anthropic principle reason that it explains why this universe has the age and the fundamental physical constants necessary to accommodate conscious life. As a result, they believe it is unremarkable that this universe has fundamental constants that happen to fall within the narrow range thought to be compatible with life

bold mine.
