Short answer: check if you can deal in ETFs with your broker. There are some ETFs/ETNs that allow you to swap a fixed amount of USD to EUR without any leverage, or alternatively with 2x leverage. For example, the ETF with ticker ERO fits your request and the following site lists other options as well:
The advantage is that the ETF should have smaller costs than a cash transaction and that you should implicitly get interest on your deposit. The latter is irrelevant at the moment given the current Fed/ECB interest rates.
In general, if you plan to hold the currency position longer (>6 months) it can make sense to buy a short duration bond ETF that invests in EUR denominated bonds. That is usually a simple and effective way to get exposure to another currency. However, given the risk of government defaults in Europe and the low interest rate at the moment I would not bother with it. Corporate bonds may be an option if you are more adventurous.
Finally, buying foreign currency (FX) as cash either physically at a bank or via a bank transfer usually involves higher costs than ETFs. Do not get fooled by banks claiming they charge "no commissions". They will use the exchange rates to hide all fees, and fees often are 5%-10% of the exchanged sum.
Cost is best measured by a round-turn: exchange USD to EUR and immediately change EUR back to USD. You get less USD back than you original had, and can calculate the costs as a percentage of the original sum. Alternatively you could compare the offered rate with the current mid price quoted on an FX site, or ask for the bid and ask price, and calculate the spread manually.
For example, at xe.com they quote me 5000$ for 3585EUR, and 3585EUR back to 4714$. The loss is roughly 5.7%, which is the spread between bid and ask. You get a better rates for larger sums and during business hours, and worse rates if the currency is not a G7 currency. The best rates I have seen with internet FX brokers were about 2-3%.
No matter what, if you deal in cash FX you also need a EUR account. Many of the big banks in the US will offer you foreign currency accounts, but they also tend to charge you fees for running it. The cheapest option is frequently to have an account locally in another country. However, banks often won't open an account for non-residents. I think there are also strict reporting and tax forms to fill out as US citizens if you open accounts abroad.