I had a quick look at the demo and it looks like the development is in the early stages- a bit of (hopefully constructive) feedback: I think they (you?) may be trying to attempt to do (and cover) too many clinical disciplines at the same time- maybe implement individual modules (like patient registration) and test them in all (old!) browsers in more detail before moving on to the next. Also think long and hard about how you implement your data model (clinical indicators e.g. blood pressure often have a context and are temporal values, how do you model these?). This is a great effort and has lots of potential.
EDIT: also, the name seems to suggest to me like there is a run on hospitals- but that may be a personal thing.
The are using PouchDB + CouchDB for offline first / syncing which is how I came to notice it (I am a maintainer for PouchDB). So their ability to sync should be reliable and well tested and if not then its bugs for us to fix :)
The epitome of "User Needs"
In the next couple of weeks when it opens I aim to join the UK governments "Digital Services Framework" where UK government software is expected to be written as open source.
The hope and intention is that development of a UK government solution (that say relies upon CouchDB) and so will have taxpayer money delivering bug fixes to users in the less well off parts of the world.
It's the right approach - and one that will have long term benefits not just for end users of software, but subtle benefits for UK / western countries.
We are getting better at this as a species - slowly. And it's attitudes displayed here that will deliver the most value in the world - even if you might not be the one to capture the cash...
As for limits, desktop browsers are generally 'unlimited' mobile browsers can be stricter and often people use wrapper projects (like cordova etc) to get round that and have unlimited storage.