What should one consider when pondering a career as a scientific editor? I'm in my 2nd year postdoc, considering getting a science-related job. I looked for jobs at PhD level, and found that one of my options is becoming an editor for a scientific journal. 

How many years' experience (after a PhD) is typical for an entry-level editor role? What else should a recent PhD know when considering such a position?
I'm a former academic publisher, and assume you're looking to join an academic publisher full-time. Do note that absent a very few elite journals like Nature and Physical Review Letters, you are not likely to be handling only journals (or be directly involved in choosing the papers to publish). You will also be doing books.


  How many years' experience (after a PhD) is typical for an entry-level editor role?


For these entry-level positions, in principle, a Bachelor's degree suffices. In fact, many of my former colleagues had more or less completely forgotten everything they learned in university and they got along just fine. Especially ironic was the case of an English Literature graduate who couldn't read calculus symbols handling an engineering journal - and she did well enough to be promoted.

(In fact there are positions even more entry-level than these entry-level positions in which a high school degree suffices, but the job responsibilities are very mundane and tedious.)

But you have a PhD. This means you won't be working at entry-level. You'll have more advanced duties that people with Bachelor's degrees might not be comfortable doing. These could include:


Acquiring new projects (this is a big one). E.g. you might be expected to identify a niche in the market for a new book, write to people who can write that book, and commission them to do it.
Attending conferences. You might have to manage the publisher's booth, answer questions about your journals and review processes, network, and yes, acquire new papers/books.
Come up with a strategy to advance your journals. This isn't easy, but it has to be you, because most people with Bachelor's degrees will have no idea. It could involve, for example, finding a trendy topic and commissioning someone to edit a special issue on it; or perhaps to split the journal into two new journals each with different focus, etc.
Help the people with Bachelor's degrees out when they approach you with (academic) problems. In my experience this was actually quite rare because most of those people aren't confident enough to ask, but it can happen. Real-life example: a paper is accepted, then during copyediting the desk editor discovers that there are only four data points. The authors have drawn a line through two of the points, but since it's only four data points, they could equally have drawn one of several other lines that also go through two points (they would fit worse, but still look good enough to be viable). What do we do now?
Make a decision on whether to accept projects. The authors put together a publication proposal, the acquisition editor has gotten reviews for the proposal, and now you have to decide whether it's a good idea to take on the project. You'll need to appraise the reviews as well as how many copies of the book you can expect to sell, etc.



  What else should a recent PhD know when considering such a position?



You're going to experience a ton of rejections when acquiring projects. I think the ratio of the number of replies to the emails sent is at best ~20% (and that's only if you're good). The ratio of acquired projects to emails sent is obviously even lower. You have to be ready to deal with this.
You're going to meet people who're very critical of you because they view academic publishing as a scam, and you by working for an academic publisher are complicit in the scam. For example, you could be doing your best to get top management to approve a lower article processing charge (APC) for your journal, knowing that these lower APCs are going to lead to the journal making a (hopefully temporary) net loss, and then get the very authors you're trying to help openly wonder how it's acceptable for you to charge APCs at all in the first place. You have to be ready to deal with this. If you end up doing the job you linked in a comment, it's going to be especially bad, because lots of academics hate Elsevier.
Don't expect to be well-paid, at least at the beginning.