The gym is built through the R&D system--I think it's one of the default options that's available once you unlock R&D through story progression. The way R&D works is that you don't choose something specific to work on initially, you just accumulate Tech, Contract, and Materials, and once you have enough you'll have the option to cash them in for an upgrade for the hotel. The gym requires 20 Tech and 15 Materials (and 0 contract). Tech is accumulated by assigning someone to R&D, and Materials are accumulated by assigning someone to exploration.
Basically, once the R&D system is open, keep assigning people to both R&D and exploration, and after a number of in-game days you'll be prompted with the option.