That means writing a test should be easy and running it should be fast (including any compilation steps). As soon as something takes more than 30 seconds, you've lost a lot of people. They've switched tabs. They're on HN or reddit or they've pulled out their phone.
You've broken the flow.
Some people can work effectively in an environment where it takes 1-10+ minutes to build something and then you run enough code to test a bunch of changes at once. You might even have multiple clients open at once and you switch between. This doesn't suit me and it doesn't suit a lot of people I've worked with.
Where does local storage fit in? Any test you write will probably need data. You don't want to mock out your data storage. You just want to use the same API and have it be backed by a hash map (or whatever) and have it easy to populate that with data.
Once you have that, local data for something interactive like a website becomes a natural extension.