If this is a possible periodic solution for a three-body problem?

What you've posted here is the classic "figure 8" orbit for a 3 body system. Originally the 3-body system was an unsolvable mathematics problem, until people started using computers to do the math for us. Recently, a set of 13 "stable" 3-body orbits have been found which were published in this paper. A diagram illustrating these orbits (with the accompanying caption) is shown below.

FIG. 1: The (translucent) shape-space sphere, with its backside also visible here. Three two-body collision points (bold red circles) - punctures in the sphere - lie on the equator. (a) The solid black line encircling the shape sphere twice is the figure-8 orbit. (b) Class I.A butterfly I orbit (I.A.1). Note the two reflection symmetry axes. (c) Class I.B moth I orbit (I.B.1) on the shape-space sphere. Note the two reflection symmetry axes. (d) Class II.B yarn orbit (II.B.1) on the shape-space sphere. Note the single-point reflection symmetry. (e) Class II.C yin-yang I orbit (II.C.2) on the shape-space sphere. Note the single-point reflection symmetry. (f) An illustration of a real space orbit, the “yin-yang II” orbit (II.C.3a).

It is hard to tell, but the gif you've posted is represented by (a) above. So I suppose the answer to your question is, yes, the figure 8 is a possible solution for the 3-body problem. However, I should caution that there is no possible 3-body system which is stable for long periods, especially if those three bodies have nearly equal masses. Our Sun/Earth/Moon system is only so stable because $M_{Sun}>>M_{Earth}>>M_{Moon}$ and they all effectively have binary orbits. For three, roughly equal mass stars, your system gets wildly unstable very quickly. Pretty much any perturbation will knock your system out.

Can anybody tell me where is the center of mass of the system?

As uhoh very nicely reasons out, the center of mass is in the center of the figure 8. Note however, that your system appears to show all bodies orbiting in a single plane. Obviously the real universe is 3 dimensions and very likely these objects would be orbiting in 3 dimensions, making the location of the center of mass more complicated. In the image above, you can see that the orbits oscillate up and down periodically, but they will do so in such a way that the center of mass is in the midplane of this oscillation, as well as the center of the figure 8.

If this is a possible periodic solution for a three-body problem?

What you've posted here is the classic "figure 8" orbit for a 3 body system. Originally the 3-body system was an unsolvable mathematics problem, until people started using computers to do the math for us. Recently, a set of 13 "stable" 3-body orbits have been found which were published in this paper. A diagram illustrating these orbits (with the accompanying caption) is shown below.

FIG. 1: The (translucent) shape-space sphere, with its backside also visible here. Three two-body collision points (bold red circles) - punctures in the sphere - lie on the equator. (a) The solid black line encircling the shape sphere twice is the figure-8 orbit. (b) Class I.A butterfly I orbit (I.A.1). Note the two reflection symmetry axes. (c) Class I.B moth I orbit (I.B.1) on the shape-space sphere. Note the two reflection symmetry axes. (d) Class II.B yarn orbit (II.B.1) on the shape-space sphere. Note the single-point reflection symmetry. (e) Class II.C yin-yang I orbit (II.C.2) on the shape-space sphere. Note the single-point reflection symmetry. (f) An illustration of a real space orbit, the “yin-yang II” orbit (II.C.3a).

It is hard to tell, but the gif you've posted is represented by (a) above. So I suppose the answer to your question is, yes, the figure 8 is a possible solution for the 3-body problem. However, I should caution that there is no possible 3-body system which is stable for long periods, especially if those three bodies have nearly equal masses. Our Sun/Earth/Moon system is only so stable because $M_{Sun}>>M_{Earth}>>M_{Moon}$ and they all effectively have binary orbits. For three, roughly equal mass stars, your system gets wildly unstable very quickly. Pretty much any perturbation will knock your system out.

Can anybody tell me where is the center of mass of the system?

As uhoh very nicely reasons out, the center of mass is in the center of the figure 8. Note however, that your system appears to show all bodies orbiting in a single plane. Obviously the real universe is 3 dimensions and very likely these objects would be orbiting in 3 dimensions, making the location of the center of mass more complicated. In the image above, you can see that the orbits oscillate up and down periodically, but they will do so in such a way that the center of mass is in the midplane of this oscillation, as well as the center of the figure 8.