Yes, you can roll your own with a ROM and limit yourself to the tiny selection on F-Droid, but you enter an increasingly niche area. While it's "technically possible", it isn't really supported well enough to justify holding it as a benefit over a competing platform from a mainstream sense.
Bear in mind, when faced with a technical question revolving around making a backup of APK files, a core Android engineer said they "didn't support piracy", which was... a pretty hefty accusation to throw around for an innocuous backup of apps for being able to sideload should they be pulled from the then-called Android Market.