I currently run a job board website, with a few blog articles that rank well.

I want to launch another job board in a similar market, and I have articles that are relevant to both sites.

I would like to publish the same articles on both sites, without having to re-write it.

The problem is, how do I deal with getting penalized for duplicate content? Is there a way to post the identical content on my second site without getting penalized?

Yes there's a very easy way to avoid duplication use canonical tags on duplicated articles, in those tag give the link of actual article, this is a way of telling Google to pass all the link juice or trust to the real article.

For Example if: https://www.example.com/blog/what-is-coding/ is your original article and you duplicatd it on some other website put below html code, with your original websites URL:

This would tell Google that you acknowledge the original source of the article, not using duplicated content so all the link juice, ranking factors may pass to the original content.

I hope this anser helps. Regards

As Hasan has pointed out, you have two options.

The first is to pick one of the websites as the one where you want the content picked up by search engines. Whichever one you publish on, you add a <rel="canonical"> meta tag to the html of both articles to indicate the primary location of the content. It's fine if it exists elsewhere as long as it has a canonical tag that points to the primary location. Everywhere the article exists should recognize the same primary location.

A second option is, instead of duplicating the content, just link from one site to the next. Since you have two sites in a similar market, you could spread some link juice around and also introduce your visitors to the other site around by cross-linking between the two sites.