You could look into Glacier for storing legacy data. There might be a way to have Lambda speak with a MySQL or Postgres database but since the functions are stateless it might have to fire up new connections on every requests which would be mad slow. You could use Amazon RDS but that could get pricey.
Honestly I’m not sure. For most of my development that I’m shipping to production I’ll use heroku and Laravel or Node.js server. The fixation with NoSQL databases in the JS community is what made me learn Laravel (PHP) in the first place so I feel ur pain!