What can I do? My new place has no Linen Closet?

Something I have done was to build shelves into the end of a closet for linens

if you have room. Usually the closet is wider than the opening leaving both ends great places to hide shelves. I just nailed short pieces of say, 2 x 2 on the inside sides and laid a board on them. Make as many as you need. :)

A shelf above the bathroom door also provides a good place for towel storage. An underbed storage box is good for linens and extra blankets.

Thank you

Sheets for each bedroom can be folded and stored on the shelf of the closet in that bedroom. A storage box or even a basket with sheets in the bottom and a pretty blanket thrown over it could work. A tall, narrow cabinet can fit just outside of or even inside of bathroom and stores towels and extra shampoo, etc. There are several "put it together" cabinets from Target, Ikea, Wayfair, etc. If you are or have a do it yourselfer, you can add a "built In" with cabinets and shelves in the space around and above the toilet.

Downsizing several times Ive learned to maximize all available storage options. Im also cursed with the lack of storage closets and like you Im lacking a linen closet too.

Purge the things you dont need or store other things that you don’t use often in another area of the house....like under the bed, garage or basement closet. Put you beds on risers and use low profile totes and spacebags under the beds. Lastly the beds with drawers below the mattress are options too.

http://www.ana-white.com/2009/11/plans-by-your-request-drawers-for-queen.html

https://www.hometalk.com/diy/s-23-insanely-clever-ways-to-eliminate-clutter-37993173?

https://www.hometalk.com/diy/s-11-storage-hacks-that-will-instantly-declutter-your-kitchen-33990625?se=fol_new-20171226-2&slg=b78c6571626002bac08964e21e0ce533-5943670&post_position=2

How about some under bed boxes on wheels

How about some under bed boxes on wheels