Persistent website with a DOI I'm writing a paper that contains links to documentation that lives as a collection of web pages. I'm looking to host these webpages somewhere such that I can create a direct link to a page from my paper, and such that the website is persistent. This means that I would like to get a DOI or something similar for the website.

Is there a service or other means so that I can accomplish this?
The PURL service (https://archive.org/services/purl/) allows you to create a persistent URL (e.g., "http://purl.org/my-thing") that redirects to a specified target URL.
You have the flexibility of changing the target URL at any time (e.g., when the target website undergoes a content reorganization or web host migration, breaking the previous URLs).
Also, partial PURL redirections are supported, for which you can use one PURL record to handle all resources that share a common location prefix (e.g., "http://purl.org/my-doc/A" redirects to "http://example.com/doc/A"). 
If the target website is yours, then you may need to shop around for a suitable web host;
free web hosting services may or may not be sufficient or appropriate, depending on the size and complexity of your website (dependencies on frameworks, databases, etc.), expected traffic, whether you mind serving ads, etc.
If the target website is third party, then you may need to monitor it over time to ensure that your PURL remains valid and accurate.
The PURL service is hosted by the Internet Archive, which IMHO will be around for a long time.

Wayback Machine (https://archive.org/web/), also hosted by the Internet Archive, is another option, especially if the webpage contents are relatively stable (or immutable).
This service allows you to "capture a web page as it appears now for use as a trusted citation in the future".
Other similar webpage archival services include 
Perma.cc (https://perma.cc/) and
archive.is (https://archive.is/).
These options work best for simple static webpages (text and images);
I recommend testing them carefully to ensure that the archived version of the website is sufficiently accurate and that you do not lose too much functionality.

figshare (https://figshare.com/) is yet another approach along the lines of open research.
This is a "repository where users can make all of their research outputs available in a citable, shareable and discoverable manner".
figshare issues a DOI for your uploaded content.
This could work if you can package the website contents suitably for viewing through figshare.

Finally, you can also directly create a new DOI for your website through a suitable DataCite (https://www.datacite.org/) member (see list of current members).



Assuming a simple static website, I would suggest the following steps:


Host your website on a free web hosting service (e.g., GitHub Pages, Neocities), create a PURL to point to it, and include the PURL in your paper and the website itself (use partial PURL redirects to handle individual pages that share a common location prefix); 
Archive a simplified snapshot of your website (perhaps on a single page) on the Wayback Machine, and include the resulting link on your website too; and
After the paper is published, add the complete paper citation with DOI to your website, and archive it again on the Wayback Machine. 


I believe this should help readers of your paper find your website and/or its contents long after the paper is published, using the published links or general search engines.