Storing medical information is tricky business. If you don't have faith in the government's My Health Record (a reasonable position to have), there are a variety of ways to store your secure data while letting others access it in case of emergencies.
You can store it locally on your computer or smartphone or sync it to your devices with a cloud storage service. Each option has its benefits and downsides, but they will all help you get at your medical information whenever you need it, and in a secure fashion.
Planned as an 'online summary of your health information' that 'can be accessed at any time by you and your healthcare providers', there are no guarantees about how your data will be used by said providers. Here's what you need to know about MHR and how to opt-out if privacy is your main concern.
Store it in a Password Manager
Security and privacy are paramount when storing medical information, so store your records where you might store other sensitive information: a password manager.
Since password managers sync and encrypt your contents across devices, you'll have your medical information wherever you are, and the ability to share it with whomever you'd like.
Store it In The Cloud
Password managers are a safe haven for delicate information, but if you want to share your records more easily, consider a secure cloud storage service. A cloud storage service like SpiderOak uses end-to-end encryption, ensuring your sensitive information is secure at every stage during its transfer.
You can create temporary, self-destructing links to share sensitive data with people. You'll have to pay, the cheapest subscription will run you $US5 ($6) per month / $US59 ($74) per year, but will net you 150GB of secure storage on an unlimited amount of devices.
If you don't have an offsite cloud storage plan (and you really should), Google's newest update to its cloud storage service Google Drive is ready to fix your gaping data backup hole by letting you pick which folders on your device you'd like to back up to Google Drive instead of forcing you to put the files into a single Google Drive folder.