Nothing in this proposed legislation (or really, just about all of the disability related civil rights laws, including the ADA) would apply to truly private facilities, including "private websites." However, if your website (or physical facility) offers what is referred to as "public accommodation" -- do you offer some service or benefit to the general public, as opposed to excluding the general public and/or not offering a service or benefit? -- then it would have to comply. Youtube would need to comply (and probably already does). Your personal website probably would not, unless (for example) you offered software you wrote or a free ebook or something like that; but even in such an example, only that part of public accommodation would be required to be made accessible.
"Making accessible" sounds like a lot of hard work, but for a website it is almost entirely based on established "best practices" for coding HTML and related technologies. Include alt attributes on your images. Specify alternate content for embedded media (descriptive text is compatible with screen readers). Use tables for tabular data, or identify the table as a layout tool not containing actual data. These are all things you should be doing anyway for a commercial/public accommodation website, given the number of folks who are browsing the Web with images turned off, Flash and other media disabled, or on devices (iPad, mobile phone) that do not support fancy sites and technologies.
Possibly the most important part is that "making accessible" doesn't mean doing away with the (theoretically) inaccessible portions. The whole point of accessibility is to provide an alternative experience that is as transparent (non-separated) as possible to the inaccessible experience. In the physical world, this often requires changes to the inaccessible experience (floor space at doors, ramps and wheelchair lifts at stairs, etc.), but in the electronic world of the Web there are established techniques (HTML elements and attributes) and technologies (screen readers, braille pads, eye-trackers) that allow persons with disabilities to access a properly coded Web page without forcing the author/creator to make a separate version for accessibility, or to do extra work that is specifically done only for accessibility purposes.