Do you have a closet that you can convert? Without more information, it's difficult to make a recommendation. You can add folding screens or curtains to hide if you want to go temporary. For permanent, you can dry wall around and add bi fold doors (louvered) doors.
A photo would save a lot of guess work for a good solution for you. Kathy gave you some suggestions that might work sight unseen. Best possibility would be to locate it near the source of plumbing for the first floor.
In addition to what is mentioned above you will also need the proper power, venting and plumping for a washer and dryer as well. Make sure that can all happen before starting any building.
Tish, please supply:
All of these things will help me to help you!
Can you hang a curtain rod?