I currently run a static website with a shop using Shopify's JavaScript SDK. All the benefits of Shopify's excellent backend and admin interface while still using a performant website that can be designed however I please.

Another option is Snipcart, which explicitely works as a drop-in shop for any website[2].

[1] https://shopify.github.io/js-buy-sdk/

[2] https://snipcart.com/