I would recommend that you rent accommodation in a compound, because Saudi Arabia's religious police don't patrol these areas. As a single woman, it's not likely you'd be able to rent outside a compound in any case.
If you want to gym in Saudi Arabia, make sure the compound you choose has its own gym, because you won't be allowed to use a regular public gym as a woman.
You may find the Women in Saudi Arabia page useful as an overview, and there is a write-up on expat compounds on the Accommodation in Saudi Arabia page.
Best of luck.
Yes, most compounds do have one-bedroom apartments available from what I can gather. But waiting lists can be long. Your company might be able to help you find something, and lots of expats are able to negotiate a package that includes accommodation organised and paid for by the company.
I found some figures from February 2015 which put the price of a one-bedroom apartment in a compound starting at 50k a year and up. 80k seemed to be a common figure, too.