If your API can run on the edge, you can get a database, function calling and user auth on Supabase's free tier. For a full Node.js runtime, Vercel's free tier is still a good option (if you don't need observability), again paired with Supabase for data.

If you can stretch "free" to "under €100 per year", then a cloud server from e.g. Hetzner will give you full control on a budget.