Q:

How can heavier elements be made from hydrogen atoms?

In a JRE podcast from a few years ago, talking about the idea of a universal constructor, Sam Harris said that we could go to a near-vacuum area of deep space, collect nothing but hydrogen atoms, build a fusion reactor, create heavier elements, and based on those elements, create a universal constructor and then build anything.
I understand that standard hydrogen atoms are made up of a proton and an electron (no neutron), while other heavier elements require neutrons.
So is it possible to create heavier elements from nothing but hydrogen? How?

A:

Saying "We" could do it is probably a bit ambitious, but stars collect hydrogen from space in gravitational collapse, and form into natural fusion reactors. The first step in stars like the Sun is the pp-n chain (diagrams from Structures of the Sky which contains more details), which creates helium from hydrogen

Variants of this process dominate in larger stars. In stars of mass greater than about 1.3 solar masses, the CNO cycle dominates. This is a catalytic process, in which protons are added to an atomic nucleus, which then decays releasing Helium.

Subsequently, Helium burning initiates the alpha ladder
 
The alpha ladder adds further helium nuclei to create a range of heavier elements, known as alpha elements whose most abundant isotopes are integer multiples of four, the mass of the alpha particle or helium nucleus. The alpha ladder starts with carbon-12 and produces
oxygen-16, neon-20, magnesium-24, silicon-28, sulphur-32, argon-36, calcium-40, titanium-44, chromium-48 terminating at iron-52 since the further fusion of alpha particles consumes energy. This is the reason for the abundance of iron in the universe. Interactions above oxygen contribute little to the energy produced by stars.
These are the main processes in stars. Other elements can be made in less common processes, and in supernovae.

