You were on to the right track; you just want to think about it in terms of what's happening at an atomic level. To get as close as possible to the actual physical diameter of a hydrogen atom, you'd want to imagine a situation where the hydrogen atoms were all physically in contact with each other. This doesn't happen in a gas, but it does happen (more or less) in a liquid.

The density of liquid hydrogen is going to vary based on external pressure, but let's start with sea level pressure just to give ourselves a benchmark. The density of liquid hydrogen is 70.85 g/L. I don't even bother doing the calculation with Avogadro's number here; I just asked Google "70.85 g/L = ? amu/nm^3" and it converted grams to atomic mass units and it converted liters to cubic nanometers just fine. The result? The density of liquid hydrogen is 42.67 amu/nm^2. We know that a single hydrogen molecule has a mass of 2 amu, so this suggests that a single hydrogen molecule in a liquid occupies a space of 0.047 cubic nanometers. Solving gives us a radius of 0.26 nanometers. So we can say confidently that a single diatomic hydrogen molecule MUST be small enough to fit within a sphere that has a diameter of 0.52 nanometers.

So how small does that make a single hydrogen atom? Well, you can look up the bond length of diatomic hydrogen and find that it is 74 picometers, or 0.074 nanometers. Covalent bond length is the distance between bonded nuclei, given intersecting electron clouds. So the diameter of a single hydrogen atom must be less than 0.45 nanometers. We're now well within an order of magnitude of the actual size.

Can we do better? Yes, we can, by looking once more at what's actually happening at an atomic level. Think about a ball pit: there's a lot of empty space between the actual individual balls. Being (essentially) spheres, hydrogen molecules don't pack into each other perfectly; they leave space in between each other. How tightly can you pack spheres together? This is a well-studied problem. In a perfectly hexagonal offset lattice, spheres can be packed with an average density (relative to the space between them) of π/3*2^0.5 or ~0.7405. That would be if the hydrogen molecules were arranged in a perfect crystalline structure. Unfortunately, hydrogen molecules are nonpolar so they don't have any intermolecular electromagnetic forces to align them that way; they will be packed randomly. This is called a random close pack and mathematicians have found that such irregular packings will produce a density of 0.64 or thereabouts.

What does this tell us? Well, if the density of liquid hydrogen is measured at 42.67 amu/nm^2, but the molecules are only packed to a volumetric density of 0.64, then 36% of that volume is going to be empty space. So the actual density of an individual hydrogen molecule is 66.7 amu/nm^2 or 0.03 cubic nanometers per molecule. This gives me a radius of 0.193 nanometers. Subtract the bond length and I get a diameter of 0.238 nanometers for a single hydrogen atom.

The actual size of an atom is known as the van der Waals radius. The van der Waals radius of a hydrogen atom is 1.09 angstroms or .109 nanometers. So the actual physical diameter of a hydrogen atom is 0.218 nanometers.

So our approach got us within 10% of the real number. Not bad!