In Unit Laundry Room Stacking Washer/Dryer for Apartment
First, congrats on the very big leap up, skipping clear over the "housemate shares and buying used stuff on Craigslist" phase of young life.  

That space is not intended for stackable separate washer/dryers, which are a recent "fad" that are a side effect of the trend to front-load washers (also a deluxe item). 

That space is for pre-built washer-dryer combos, which are simply a washer and dryer pre-stacked onto one chassis, like this unit. These units are a stock-and-trade of rental units and condos.  Most are small capacity to make them narrow; large capacity units can be had. 



Note the wedge opening that allows access to the top loading washer on the bottom...

Lately, however, the new trend is for "washer-dryer all-in-one" units, which seem to defy the laws of physics! They place washer and dryer function in a unit the form-factor of a washer.  Put clothes in dry, they come out washed and dry.  All the rage on tiny houses, boats, etc. I'm skittish of this one because of its low price, but there you go...



Dirty clothes go in, dry clothes come out...

Keep in mind ordinary wear and tear should not be counted against you, as a result many tenants get their entire deposit back. But you do not have any legal right to modify your apartment, and you will pay the full cost of restoring your apartment to status quo ante, which will be billed to you if the deposit doesn't cover it.