Eunice here from Shopify, thanks for posting!
Dogecoin and other cryptocurrencies are not able to be selected as your store currency. The workaround for this is to use a currency converter app such as BEST Currency Converter by Grizzly Apps SRL. A currency converter app will allow your customers to view your product prices in their selected currency.
I noted the BEST Currency Converter app works with Bitcoin as a currency, however, there was no mention in the app listing for Dogecoin. I recommend getting in touch with the App Developer directly to see if this is available, or if it can be made available. Using this currency in your Shopify store is still relatively new, so we are expected to see more and more apps cater to this in the near future. You can find the contact details for the App Developer in the app listing:
If you'd like to see this become a Shopify feature in the future then you can also submit a Shopify feature request by logging into the Shopify Help Centre and selecting chat, callback, or email from the support options. We'd love to hear from you!