Among the many Android N features revealed at Google IO was the promise of seamless updates, allowing your device to install updates in the background without rendering it temporarily unusable, but this is one feature that isn't likely to arrive on any current hardware.
Speaking to Google, Android Police learned that while it would technically be possible to achieve on some current handsets, such as the Nexus 5X or Nexus 6P, it would mean repartitioning the entire phone, so that there were two partitions live at once - one to download and install updates onto and the other to continue using the phone from.
But repartitioning isn't easy - it would require hooking your handset up to a computer and, more worryingly, risks breaking your phone or losing data - so it's both hassle and, frankly, not worth the risk from Google's perspective.
New phone needed
It's possible that some enterprising users will develop tools to make it work, but if so they will likely be entirely unofficial and used entirely at your own risk. Otherwise, if you really can't bear to look at another "Android is updating" screen, you'll to have to buy a phone that runs Android N or above out of the box, the first of which will likely be the new Nexus devices we're expecting later this year.
Everyone else will just have to continue finding ways to fill that phoneless time. We'd suggest using it to learn something new, but without our Duolingo and Skillshare apps we'd probably just end up staring into the abyss until our phone works again.