Practically, this means no US cloud hosts. I'd recommend replicating your cluster to a European cloud provider if you want to be sure. Hosted Elasticsearch and MySQL/Postgres are available in tons of European cloud providers, sometimes for a lower price than their American competitors. It's more overhead for sure, but nothing business-ending.
However, the GDPR only protects personally identifiable data. A lot of data is PII, but not all data is PII. You might not need to bother if you don't collect anything that's unique to a person (though user accounts might pose a problem even if you don't process any other PII).
Depending on the size and turnover of your business, you may also need to comply with some other GDPR requirements (privacy officer etc.) but that's usually nothing more than appointing someone within your company to deal with+take into consideration privacy concerns (something your company should be doing anyway if it's ethically managed). Your data storage will probably be a bigger problem for your business.