Do not take this too literally. It is a mathematical approximation. Actually, the assumption underlying is not that gas "particles" are points so much as that the volume occupied by gas particles is much smaller than the volume occupied by the overall gas. In the most extreme possible extrapolation of this assumption, gas particles would have no volume at all. Of course, this is not true, but the idea is that if a gas is dilute enough / low enough pressure, the volume actually occupied by atoms in a gas so small compared to the total volume available for gas atoms to explore, that the gas atoms can be approximated as basically having no volume. This makes determining quantities like how far atoms travel before hitting each other on average and the total free volume of the container much easier to calculate. As you know, though, atoms do actually have volume, and so at very high pressures, when the amount of space taken up by atoms relative to the total amount of volume in their container is no longer negligible, the determination of how long atoms can travel before colliding with each other and the total volume of free space can no longer neglect atomic volume without making noticeable errors. This is one reason the ideal gas law breaks down at very high pressure.

Consider an analogy. Suppose you want to know how much empty space there is in the solar system. You know that the solar system has planets, so it's no all empty space. The planets in our solar system orbit the sun in a vast amount of space. The planets are big, but compared to the total volume available in the solar system, they are ridiculously small. The volume of the earth (diameter ~ 6400 km) is about 130,000,000 km^{3}. The diameter of the solar system is on the order of 300,000,000,000 (300 billion) kilometers, meaning it has a volume of 3x10^{23} km^{3}. So, the volume of the solar system is on the order of 10^{15} times larger than the volume of the earth. Of course there are other planets and also the sun, but you get the idea. In determining how much empty space is in the solar system, you don't really need to subtract out the volume of planets. On the scale of the solar system, it would be appropriate to mathematically treat the planets as basically having no volume at all - whether you include planets or not, you get about the same volume of empty space in the solar system. But, if you squished billions and billions of more planets in the solar system, you could no longer neglect the volume that the planets take up. Your calculation assuming no planetary volume would be way off. Plus all life on earth would probably end. :/