It’s just that atoms are even smaller. The nucleus etc. are all composed of atoms.

Think of the smallest thing you can think of. Well, atoms are much,much,much smaller than that. That’s why they all fit in. It will much more than 100,000s of atoms in a cell.

and rest assured there is no hyperbole or exaggeration in gmander’s answer. Actually I’m reading science forums now and they all seem convinced there are around 192 trillion atoms per cell in the human body.

Here is a Visual comparison of various distances .

A typical cell is 10m to 100 um (microns) across.
A typical atom is 0.0001 um.
A water molecule is 0.0003 um.

The atoms are everywhere. They just too small to see. .

Here are a couple of computer-generated animations of cellular activity at the nano-scale, where atoms are represented as the smallest little bumps composing the molecules. These are great visualizations that bridge the gap between the ultra-microscopic and the atomic:
Here and Here.

These should help put the scales in perspective. Note that the features depicted are way too small to be seen in a light microscope, but can be made out with electron microscopy.

Everything the cell is made up from is made of atoms. There literally is no form of matter on the earth is isn’t composed of atoms (or ionized atoms and free electrons in the case of plasma (not to be confused with blood plasma) which you’d find in plasma TVs and neon signs).

… [T]here are ~10^14 atoms in an ordinary human cell, compared to ~5×10^11 stars in the Milky Way. So each cell in the human body is made, quite literally, of a galaxy of atoms. And by a remarkable numerical coincidence, the number of cells in the human body is about the same as the number of atoms in a cell. Thus every human being can invoke Walt Whitman in saying “I am large, I contain multitudes.” Exploring this biological galaxy-of-galaxies is the 21st Century’s new frontier of biology and medicine.
http://courses.washington.edu/goodall/MRFM/NIH_nanocenter.html

10^6 = one million
10^9 = one billion
10^12 = one trillion

This question is a bit like asking how they fit all the water into the ocean. The ocean is made of water, and the cell is made of atoms.

Most atoms are actually part of molecules, and some of these molecules are quite large. The best example is a DNA molecule with a width of about 2.5 nanometers. The length of the DNA molecule of human chromosome number 1 67,000 nanometers or 67 microns.