Yes. There are both washers and dryers you can use in apartments without hookups. The dryer will need a window for the exhaust, or if it's cold, just let it blow out directly into your apartment from some bonus heating. Here's one. For the washer you need to be able to put it close to a sink or bathtub. I had one that came with an adapter for the sink. Something like this.
Just got to dry your pants one at a time
Panda 110V Electric Portable Compact Laundry Clothes Dryer, 1.5 cu.ft, Stainless Steel Drum Black and White, PAN725SF https://www.amazon.com/dp/B00EAY540S/
So there are 2 options that I know of that would work with a standard outlet, they are on the opposite of the cost spectrum. There's nothing really in between that works on a standard 110v.
one is a "portable" vented dryer. Example below, but a fair amount of companies make these.
https://www.amazon.com/dp/B00EAY540S/ref=cm_sw_r_cp_apa_glc_fabc_1J0KGFGH1460SBJ6BX7F
The other is a 110v heat pump dryer that plugs into a standard wall outlet. It's ventless, so you'd cap your vent and have the drain hose from the unit share the drain with the washer. This isn't against code to have a ventless unit.
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Dryer is a Panda 110V Electric Portable Compact Laundry Clothes Dryer, 1.5 cu.ft Works fantastic and does the job just as well as the full size dryers in my community laundry. It will do just about anything except large bedding. I have not tried a heavy winter coat because I am in Southern California and don't use one but I did just wash a regular jacket and a load of sheets/pillowcases and it worked perfectly fine. Likewise with the washer.
Washer: SUPER DEAL Portable Compact Mini Twin Tub Washing Machine 13lbs Capacity. I use an extension cord because of the configuration of where my bathroom is versus my power outlet leaves me no choice, but that might not be an issue for you depending on the layout of your place. I run the washer in my bathtub so the water can just drain, and I use my extended shower head to fill the basin. It comes with a tube you can use to fill it but I find it is much more effective and faster to do it with the shower head.
Things to know: The washing capacity is smaller than a regular washer for obvious reasons and for some reason the spinner is smaller than the wash side. I just split my clothes into two spin cycles and it is no big deal. When I got the washer it worked great and then about 4 loads in the spinner stopped working. Thought I was going to have to return it at first, but it turned out that all I needed to do was unplug and replug it into the power outlet. Occasionally it will do that same thing again and as soon as I redo the connections, it is fine. It works great 99.9% of the time. I think that just happens because I am using the extension cord. However, I figured I'd let you know so you are aware so you know how to troubleshoot if it happens, because the manual they include is really not helpful at all, lol. Good luck with your new rental. Hope you find an awesome place.