> Has anyone described or catalogued all sets of non-negative real numbers that are closed under addition?

In recent years, there has been a lot of work on the arithmetic of *Puiseux monoids*, that is, submonoids of the positive cone of the additive group of a totally ordered field $K$: The focus has been mostly on _rational_ Puiseux monoids, where $K$ is the rational field (with its usual ordering); but there are also a few papers about the general case. For further details, I can only recommend to have a look at the work of Felix Gotti et al., starting with

   - S.T. Chapman, F. Gotti, and M. Gotti, *When Is a Puiseux Monoid Atomic?*, 
Amer. Math. Monthly 128:4 (2021), 302-321.

This article is a pleasure to read and, together with 

   - S.T. Chapman, F. Gotti, and M. Gotti, *Factorization invariants of Puiseux monoids generated by geometric sequences*, Commun. Algebra 48:1 (2020), 380-396,

it offers a lucid introduction to (different aspects of) the classical theory of factorization. 

From the reading of these papers, it will become clear that even a classification of rational Puiseux monoids is more or less hopeless. In particular, you may want to have a look at

  - A. Geroldinger, F. Gotti, and S. Tringali, *On strongly primary monoids, with a focus on Puiseux monoids*, J. Algebra 567:1 (2021), 310-345.