Do e4s get bah in Germany?

If you are a single E1 to E4 you will likely live in the barracks. If you are married, you will likely live in family quarters.

Overseas there is no BAH, there is Overseas Housing Allowance (OHA). OHA tends to go to single officers and some single NCOs. Things change from year to year.

Your 1SG and housing staff will address availability.

BAH is only for your dependents when you're overseas. Assuming you are a single Soldier, if you live in the barracks you don't have a housing allowance. When you move out, if you move out, you receive OHA. OHA is based on the max your rent is eligible for as opposed to BAH which is what you're given for that area regardless of your rent.
You will also receive Cola

OHA (Overseas Housing Allowance) is determined by where you’re stationed (like BAH). You will only receive the amount stated in your rental/lease agreement. Unlike BAH, there won’t be anything “left over” if you find a cheap rent. You will only get OHA if there is no suitable/acceptable housing available on post. The housing office in Graf will be able to tell you about programs available to help with the utilities. This is all assuming you are married. If you are “without dependents” and Junior Enlisted, you will be living in the barracks.