Can I play/practice Fortnite's Battle Royale locally?

This will eventually be supported as part of Fortnite Battle Royale's Custom Matchmaking feature, which is currently in private testing.

Official reddit post and the same post on the Epic Games forums:

We are re-enabling the "Custom Matchmaking" button on PC. Why?

To test private matches. PC players will be able to see the button, but it cannot be used by anyone outside our private tests. We are going to be leaving the button on as we continue to test the private match feature.

It has been anounced that in patch 4.3 Fortnite will be adding a practice mode (called The Playground).

See http://uk.businessinsider.com/fortnite-practice-mode-2018-5

And the fortnite blog: https://www.epicgames.com/fortnite/en-US/news/state-of-development-v5

The Playground LTM will load you into the Battle Royale Map with some adjusted settings.

Battle and build to your heart's content with an extended period of time to roam around the map as well as increased resource generation. All treasure chests and ammo crates will be spawned, try droppin’ in different spots and scope out the loot. Friendly fire is on so you can scrimmage with your squad (up to 4 friends per match), but fear not you’ll respawn immediately.