Does accessing my network storage (NAS, hard drive) affect my bandwidth? For example transferring stuff from and to my NAS, does that take away the 250gb that comcast gives me?
If your NAS is set up properly, the short answer concerning your ISP "bandwidth" is: no. Though technically, you're not asking about bandwidth over your network, but are asking about your ISP band caps.
Unless you've set your NAS to download things like torrents or you're accessing it outside of your home than like Velin said the answer is no. Transferring a file from your computer to your NAS locally doesn't touch your ISP bandwidth.