Atoms are small units of matter, but they are made up of smaller particles: electrons, neutrons, and protons. The protons and neutrons are actually made up of even smaller particles, quarks. In our current understanding of physics, electrons, quarks, and other elementary particles are the smallest known units of matter/energy. It is possible that there might be even smaller units that are compose these elementary particles, which might help to explain and understand certain patterns observed in their properties, but these have yet to be discovered. Even some potential elementary particles (such as the graviton) remain hypothetical.
In an earlier post, I discuss how hopelessly large these numbers are, even if we could manage to store information at a subatomic scale:
As @jlt noted, the storage of information in a computer requires more than just energy, but also the physical structure of the circuits. The densest electronic storage of information comes from flash memory, which stores information by essentially charging energy into tiny electronic devices. However, storing each bit requires a separate device that is roughly on the scale of nanometers, and a cubic nanometer of material contains, roughly, quadrillions of atoms. Then, any practical use and manipulation of the stored information requires surrounding circuity and packaging, along with the rest of the computer system, which is many orders of magnitudes larger.
Beyond electronics, the highest density information storage seems to be an experimental DNA-based system, which seems to uses dozens of atoms per bit of information stored. From Density (computer storage) - Wikipedia
In 2012, DNA was successfully used as an experimental data storage medium, but required a DNA synthesizer and DNA microchips for the transcoding. As of 2012, DNA holds the record for highest-density storage medium.[12] In March 2017, scientists at Columbia University and the New York Genome Center published a method known as DNA Fountain which allows perfect retrieval of information from a density of 215 petabytes per gram of DNA, 85% of the theoretical limit.[13][14]