Slight frame challenge here: You don't need an account denominated in Euros in order to spend money electronically in Europe without fees.
Many U.S.-issued credit cards, especially those that are focused on travel, charge no foreign transaction fees. I have at least half a dozen of them myself. They're very common in the U.S., including some that do not charge annual fees. Just tell the merchant to charge your card in Euros and your bank will automatically convert USD to Euros, typically at something very close to the current inter-bank rate.
Additionally, if you want Euros (or any other currency) in cash, you can get a checking account with a debit card with a U.S. bank in USD that does not charge fees for foreign transactions and which also reimburses ATM fees (including those charged by the owner of the ATM.) Then, as with using credit cards at merchants, just tell the ATM to withdraw from your account in whatever the local currency is (i.e. Euros in the Eurozone) and your bank will exchange it with USD at the current exchange rate and also reimburse you whatever fees the ATM may charge. Personally, I use a Schwab checking account for this purpose, though there are other similar accounts available. The Schwab checking account is completely free and charges no fees whatsoever to withdraw foreign currency, so I can just go to any ATM and withdraw the local currency. Schwab then credits whatever fees the ATM owner may charge for the transaction back to my account.
This is all far easier than maintaining a Euro-denominated account and works just as well wherever you travel, not just Europe. Your accounts all remain denominated in USD and merchants just charge them in whatever their local currency is. Just make sure to tell the merchant, ATM, etc. to charge your account in the local currency and let your bank do the conversion for free rather than having the merchant do it. There is no reason whatsoever to pay the merchant to charge your account in USD. They almost always give you very unfavorable exchange rates.