What does it take to get a feature request implemented?
First, you need to convince Stack Exchange that the feature is actually something that should be done.
Being popular is not enough - we are aware that Meta users are not, in fact, representative of all of the users and that sometimes, something that looks like a good idea at first blush is, in fact, a really bad one.
Now, some requests are all to do with the client side - things that user style sheets and user scripts can handle - and there are a bunch of those around (take a look at stackapps). These are the kinds of features that you can implement yourself.
As for other features - we have written and open sourced a bunch of libraries that we use in our codebase. If you want to work on features of those - they are listed on stackexchange.github.io.
The Stack Exchange codebase itself, outside of the OSS we publish, is closed source.
Is it possible for me to request to be allowed to do so?
Only if you get hired as a developer by Stack Overflow.