I found one place that does storage in Santo Domingo called International Packers. They have an office on Lincoln and a storage facility on the outside of Santo Domingo. This is certainly an option since they offer insurance and drop off/ pick up. I was hoping that others on this site could offer other options or places they have used in the past.
I think you are right in that there aren't any places like in the USA such as the company Public Storage.
Thanks for the info. I couldn't find almadom, but was able to contact Almadeca. Almadeca does have storage, but they don't offer the same pickup and delivery services like international packers. I will continue to call and see what I can find. Again, thanks.