Travel Trailer Storage

Dont know if you specifically want something in the city, if not, most of the state parks around Kerr Reservoir have private covered RV and boat storage locations near their entrances. Short ride up to Henderson should give you several decent options.
I'm sure Falls lake has some around it too.

I'm looking for storage either in Raleigh, NC, or close to Raleigh, NC. Preferably in Raleigh, NC.

Did you google RV storage near me?

Look on Craigslist

You can certainly find storage facilities in Raliegh.
Expect to pay double or triple what it cost 20 miles outside the city.
I suggested the cost savings options, but feel free to splurge.
Enjoy.