I've heard from renowned astrophysicists that we don't yet know whether or not the Universe is infinite. How is that possible regarding the big bang theory is accepted (as they all do)? Are they referring to the existence of other Universes when they say it could be infinite, or what?
What I mean is, even if it is capable of expanding, if everything originated at the big bang how can that space turn to be infinite in a finite time - the age of the universe.
In the standard ΛCDM model of the Big Bang, the universe is infinite and has always been such. The Big Bang singularity happened everywhere, in the sense that far back enough in time, the density diverges to infinity at every place.
But this is just a particular model--it assumes that the universe if spatially flat and is globally homogeneous and isotropic. There are extended models in which it is not exactly flat, and so could be finite even if it is still homogeneous and isotropic (if the curvature is even slightly positive). And of course we don't actually know whether it is homogeneous and isotropic at scales much larger than we actually see. Some inflationary models imply that it isn't.
To clarify: the ΛCDM model uses the assumes a spatially flat FLRW solution of general relativity, in which space is the Euclidean $3$-space The Euclidean $3$-space is the only flat homogeneous and isotropic $3$-manifold, so there no way to make it finite without violating at least one of those modeling assumption (e.g., a flat torus could have the same form for the metric, but would not be globally isotropic).
I think the source of confusion between the two concepts - the Big Bang singularity and an infinite universe - is the misconception that the universe began as a finite expanse originally. This misconception easily arises from analogies using present-day logic and numbers that were not applicable in the early universe. For example, I've heard it said that shortly after the Big Bang, the entire observable universe was the size of a grapefruit, but that explanation neglects to mention that grapefruits would have been much larger then.
The problem is that space is where we can measure how large something is, but space expands, so something that is a certain distance away currently was a lot closer a long time ago, even if neither object has moved in the normal sense. As an analogy to help illustrate the effect:
You and I are standing on a preposterously large deflated balloon. You set down a meter stick, make a mark on the balloon at each end and we each stand on one mark and are now a meter apart. Then I turn on a pump and start inflating the balloon. As the balloon inflates, the surface stretches out and you and I appear to get farther from each other, when though we're not 'moving' (e.g. walking away from each other): now we have conflicting sets of information to consider; according to the marks on the balloon surface we're still one meter apart, but according to the meter stick in your hand (which is not expanding) the distance is greater than that.
Note that while I called the balloon "preposterously large," it could have been infinitely large and still behave the same way. I point this out because I've seen in comments on other answers that you don't see how space could be both infinite and expanding - that if it's expanding, then it must have been previously finite. That is incorrect: in fact, because infinity is the quality of unboundedness, something that is infinitely large can always get bigger, because by definition there is no upper bound on its size.
Note also that if you recorded the earlier analogy in reverse, it would appear that space was shrinking such that a several-meters distance between us reduced over time to one meter. If you continue shrinking the universe in such a manner, it eventually becomes the case that there is zero distance between us. And if you apply that to a scenario where there are people infinitely distributed across the balloon, all of them would come closer together as the balloon deflated, until there was zero distance between any two people... in theory, at least, since real human beings have size. Energy and space don't have size, however, so at the point of the Big Bang, space was still infinite (since an infinite/unbounded space cannot shrink to become finite/bounded) but the distance between any two points in space was zero.
So if you could go back in time to the Big Bang you'd see an infinite ocean of energy, since all the energy was "shoulder-to-shoulder" (infinitely dense) but it rapidly expands (and therefore cools) to the point that basic particles can form, then later matter and molecules. Of course since your size would depend on the metric of space, it wouldn't necessarily look like space was expanding, but simply like the energy and matter were cooling down. In fact we still see this as an effect of spatial expansion in the redshift of light from distant sources: the light "cools down" or loses energy along the way because it is stretched out on its journey through space.
"How" can the universe be infinite? So I am inferring that there must be a 'procedure' of sorts to make it infinite.
Space can exist by itself without having anything. Vacuum, Intergalactic space, the huge voids seen in the large scale structure maps are examples.
This is means no matter how much the universe expands, space cannot break apart like atoms. One may say it has no elastic modulus. One may use this as a basic premise to explain how there can be an infinite universe.
Alternatively, theories of cosmic topology are in good agreement with the predictions of our universe's matter density being very close to 1 which points to a universe expanding forever. (see http://www.scholarpedia.org/article/Cosmic_Topology#fabre:2013)
It cannot be infinite in a finite time because time is space and time are more or less the same. They are one entity called spacetime.
It would be cool if Hubble saw a signpost marked: "please do not go beyond this point, this is the edge of the universe." perhaps even a wall or a balustrade.
How can the universe be infinite? infinite size, infinite age, infinitely small quantum constructs? The main problem we have with infinity is that it defies human understanding.
Spherical and circular equations pervade the universe: Stars, CMB, planet and moon orbits, atoms, photons, Pi, they all can use sine-cosine equations. Rotations are infinite. Pi and sine-cosine are infinite.
If the cosmos is made up of infinite maths functions, why would it not be infinite?
How many angles does space have? infinity. how long does a galaxy turn around? eternity. The circles of angular momentum go on ad-eternum. rotations are infinite.
Why would there be a spacial limit to infinitely large or infinitely small space? Because it defies human understanding? that's not a good enough reason.
Everything in the known universe depends on mathematical principles which are infinite, numbers are infinite, angles are infinite, generally sine-cosine, linear, circular and spherical objects have infinite turns.
The mandelbrot set is infinite. It's a much simpler equation than the objects of the universe, and it is infinite. So, i'd encourage you to study the mandelbrot and ask yourself: how can it be infinite? Then you will have some advantage to apply that study to objects of the universe.
There is a curious thing that bridges the infinitely large and small objects in the universe: Black hole singularities are nearly entirely composed of sine-cosine maths descriptions, but they withhold millions of suns of atoms in the space the size a money coin or a CD or a nano-meter, all inside a sine-cosine disk, which is a circular phenomenon weighing billions of stars. If suns can fit into basketballs, and perhaps atoms, how can we judge what is big or small at all? If very big things are reborn to very small, and grow to very big again, it's kind of like the circle of life, and it means that you are the center of the universe, which is a good feeling. Atoms and Stars don't matter compared to you.