the limit is 5MB in general i think. For the chrome browser you can increase that limit programmatically till 30 or 50mb ... something like that. but thats not recommended.
It depends a little bit what do you want to achieve. if you need synchronisation with your backend db (for offline usage) i would use pouchdb. If you only want to store data (many data sets) you can use sqlite.
If you are storing only some flags and so on lokijs is a good memdb with the possibility to store all the data to a json file --> and the good point per default it uses localstorage --> so if you ever want to run your code in browser for testing or production purpose -> it is working 